Errors in Numerical Solutions of Shock
Physics Problems
A Dissertation Presented
by
Yan Yu
to
The Graduate School in Partial Fulfillment of the Requirements for the
Degree of
Doctor of Philosophy
in
Applied Mathematics and Statistics
Stony Brook University
December 2004
Copyright c© byYan Yu2004
Stony Brook University
The Graduate School
Yan Yu
We, the dissertation committee for the above candidate for the Doctor ofPhilosophy degree, hereby recommend acceptance of this dissertation.
James GlimmAdvisor
Department of Applied Mathematics and Statistics
Xiaolin LiChairman
Department of Applied Mathematics and Statistics
Yongmin ZhangMember
Department of Applied Mathematics and Statistics
Roman SamulyakOutside Member
Brookhaven National LaboratoryCenter for Data Intensive Computing
This dissertation is accepted by the Graduate School.
Graduate School
ii
Abstract of the Dissertation
Errors in Numerical Solutions of ShockPhysics Problems
by
Yan Yu
Doctor of Philosophy
in
Applied Mathematics and Statistics
Stony Brook University
2004
Advisor: James Glimm
We seek error models for shock physics simulations which are robust and
understandable. We propose statistical models of uncertainty and error in
numerical solutions. To represent errors efficiently in shock physics simulations
we propose a composition law. The law allows us to estimate errors in the
solutions of composite problems in terms of the errors from simpler ones. We
formulate and validate this composition law for shock interactions in planar
geometry. We also explore complications introduced by spherical flow in the
analysis of errors in the numerical solutions. We illustrate that idea in a very
simple context.
iii
For shock interactions in spherical geometry, we conduct a detailed anal-
ysis of the errors. One of our goals is to understand the relative magnitude of
the input uncertainty vs. the errors created within the numerical solution. In
more detail, we wish to understand the contribution of each wave interaction
to the errors observed at the end of the simulation.
Key Words: uncertainty quantification, error model, composition law,
Riemann problem.
iv
To my parents and my loving husband
Table of Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . xix
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 FronT ier Front Tracking Algorithm . . . . . . . . . . . . . . 7
1.2.1 Interface propagation . . . . . . . . . . . . . . . . . . . 7
1.2.2 Interior computations . . . . . . . . . . . . . . . . . . . 10
1.3 The Wave Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Dissertation Organization . . . . . . . . . . . . . . . . . . . . 14
2 Error Analysis for Planar Geometry . . . . . . . . . . . . . . 16
2.1 Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 The Statistical Numerical Riemann Problem . . . . . . . . . . 17
2.2.1 The Isolated Shock and Contact Wave . . . . . . . . . 19
2.2.2 Shock Contact Interactions . . . . . . . . . . . . . . . . 23
vi
2.2.3 Shock Crossing Shock Interactions . . . . . . . . . . . . 28
2.2.4 The Contact Reshock Interactions . . . . . . . . . . . . 29
2.3 The Composition Law . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1 A Multipath Integral for a Nonlinear Multiscattering
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.2 Evaluation of the Multipath Integral . . . . . . . . . . 33
2.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.1 Errors in Fully Resolved Calculations . . . . . . . . . . 35
2.4.2 Errors in Under Resolved Calculations . . . . . . . . . 38
3 Error Analysis for Spherical Geometry . . . . . . . . . . . . 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 The Statistical Numerical Riemann Problem . . . . . . . . . . 43
3.2.1 The Single Propagating Wave . . . . . . . . . . . . . . 44
3.2.2 The Shock Contact Interaction . . . . . . . . . . . . . 47
3.2.3 Shock Reflection at the Origin . . . . . . . . . . . . . . 51
3.2.4 The Contact Reshock Interaction . . . . . . . . . . . . 52
3.3 Composite Shock Interaction Problems . . . . . . . . . . . . . 54
3.4 Error Decomposition . . . . . . . . . . . . . . . . . . . . . . . 55
4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1 2D Shock Wave Interactions in Perturbed Spherical Geometries 64
5.1.1 Single Mode Perturbed Interface . . . . . . . . . . . . . 65
5.1.2 Chaotic Mixing . . . . . . . . . . . . . . . . . . . . . . 66
vii
6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.1 Complete List of Ten Riemann Problems (Planar Geometry) . 69
6.2 Errors in Resolved Calculations . . . . . . . . . . . . . . . . . 81
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
viii
List of Figures
1.1 Flow chart for the front tracking computation. With the ex-
ception of the i/o and the sweep and communication of interior
points, all solution steps indicated here are specific to the front
tracking algorithm itself. . . . . . . . . . . . . . . . . . . . . . 8
1.2 Schematic diagram illustrating the operation of a wave filter. Left:
computational data (squares) are fit to an error function. The er-
ror function depends on four parameters, a position, a width, and
two asymptotic values. These determine the wave position, width
and height, with subgrid accuracy. Right: a piecewise linear con-
struction is fit to the rarefaction or compression wave data. . . 13
2.1 Left. Space time density contour plots for the multiple wave inter-
action problem studied in this section. Right. Pressure contour
plots for the base case considered here. . . . . . . . . . . . . . . 18
2.2 Left: Type and location of waves as determined by our wave
filter analysis for the base case considered here. Right: Schematic
representation of the waves and the interactions, with labels for
the interactions, taken from the left frame. . . . . . . . . . . . 19
ix
2.3 Ensemble mean shock width and the standard deviation of the
shock width (left frame). The mean width, equal to about 2∆x,
is much larger than the standard deviation, indicating that the
mean width is essentially a deterministic feature of the solution.
Convergence properties of the travelling wave to the steady state
values on each side of the wave (right frame). The straight line in
the right frame is the asymptote to the exponential convergence
rate, with slope 0.01 in units of ∆x. . . . . . . . . . . . . . . . 20
2.4 Ensemble mean contact width for isolated noninteracting waves.
Because the width is entirely grid related, we record width in units
of ∆x and time in units of the number of time steps. The standard
deviations are also plotted, and are the points to the extreme left
in each frame. Left (step down): we observe an increase from 2
cells to 30 over 104 steps and an asymptotic growth rate cct1/3,
where cc ∼ 1 depends on the flow Mach number. The straight
line in the left frame is the asymptote to the contact width, with
slope 3. Right (step up): We observe a bound on the contact
width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Left: ensemble mean shock and contact widths before and after
a shock contact (step up) interaction and standard deviations.
Right: ensemble mean shock and contact position errors as a
function of time, expressed in grid units. Step up case. . . . . . 24
x
2.6 The solution and its errors at the point (x, t) can be obtained
by “adding up” the solution and errors for the waves within the
domain of dependence . . . . . . . . . . . . . . . . . . . . . . . 32
3.1 Left: space time density contour plot for the multiple wave in-
teraction problem studied in this chapter, in spherical geometry.
Right: type and location of waves determined by the wave filter
analysis with labels for the interactions. Here I.R. denotes the
inward moving rarefaction. . . . . . . . . . . . . . . . . . . . . . 43
3.2 Left. Mach number vs. radius for a single inward propagating
shock. Right. The same data plotted on a log-log scale. . . . . . 45
3.3 Left. Mach number vs. radius for an outward moving shock wave
starting at different radii r0. Right. The same data plotted on
a log-log scale; the dashed lines in this plot represent the power
law model (3.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Ensemble mean contact width for a single propagating contact.
We record the width in units of ∆x. The standard deviation is
also plotted, as the points to the extreme left. . . . . . . . . . . 47
3.5 Left: ensemble mean inward/outward moving shock and contact
widths after a shock contact interaction. Right: ensemble mean
shock and contact position errors as a function of time, expressed
in grid units. The associated standard deviations are extremely
small, not shown in the plots. In the legend, C. denotes the
contact while I.S. and O.S. are the inward and outward moving
shocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
xi
3.6 Schematic graph showing all six wave interaction contributions
to the errors or uncertainty in the output from a single Riemann
solution, namely the reshock interaction (numbered 3 in the right
frame of Fig. 3.1) of the reflected shock from the origin as it
crosses the contact. The numbers labeling the circles refer to
the Riemann interactions contributing to the error. The numbers
labeling the line segments refer to the different error propagating
paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.7 Pie charts showing the contribution of each wave interaction dia-
gram to the error variance of the wave strength at the output of
interaction 3, for a solution using 500 mesh units. . . . . . . . . 59
3.8 Pie charts showing the contribution of each wave interaction dia-
gram to the error variance of the wave strength at the output of
interaction 3, for a solution using 100 mesh units. . . . . . . . . 60
5.1 Density plot for a spherical implosion simulation with a perturbed
interface (single mode). The grid size is 200 × 200. . . . . . . . 66
5.2 Density plot for a spherical implosion simulation with a chaotic
perturbed interface (multiple modes). The grid size is 200 × 200. 67
6.1 Problem 1: Shock-contact (step up) . . . . . . . . . . . . . . . . 70
6.2 Problem 2: Shock-wall interaction . . . . . . . . . . . . . . . . . 70
6.3 Problem 3: Contact-shock (step down) . . . . . . . . . . . . . . 71
6.4 Problem 4: Rarefaction-wall . . . . . . . . . . . . . . . . . . . . 72
6.5 Problem 5: Contact-rarefaction . . . . . . . . . . . . . . . . . . 73
xii
6.6 Problem 6: Shock-shock overtake (two waves of the same family) 74
6.7 Because the width is entirely grid related, we record width in
units of ∆x and time in units of the number of time steps. . . . 76
6.8 Problem 7: Compression-wall . . . . . . . . . . . . . . . . . . . 77
6.9 Problem 8: Contact-compression . . . . . . . . . . . . . . . . . . 79
6.10 Problem 9: Rarefaction-wall . . . . . . . . . . . . . . . . . . . . 80
6.11 Problem 10: Contact-rarefaction . . . . . . . . . . . . . . . . . . 81
xiii
List of Tables
2.1 Parameters that define the base case for the shock contact inter-
action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 The SNRP shock contact (step up) interaction. Expansion coeffi-
cients for output wave strengths, wave widths and wave position
errors (linear model) for input variation ±10%. Here the base
case input contact wave width is zero. . . . . . . . . . . . . . . . 26
2.3 The SNRP defined by the crossing of two shocks. Expansion
coefficients for output wave strengths, widths and position errors
(linear model) for input variation ±10%. . . . . . . . . . . . . . 28
2.4 The SNRP shock contact (step down) interaction. Expansion
coefficients for output wave strengths, widths and position errors
(linear model) for input variation ±10%. . . . . . . . . . . . . . 30
2.5 Predicted and simulated errors for output wave strengths, wave
widths and wave positions, Interaction 1. . . . . . . . . . . . . . 36
2.6 Predicted and simulated errors for output wave strengths, wave
widths and wave positions, Interaction 2. . . . . . . . . . . . . . 36
2.7 Predicted and simulated errors for output wave strengths, wave
widths and wave positions, Interaction 3. . . . . . . . . . . . . . 37
xiv
2.8 Case 1. The contact-shock interaction (step up). Errors for out-
put wave strengths, wave widths and wave position. Comparison
of under resolved simulation and prediction. . . . . . . . . . . . 39
2.9 Case 2. The shock crossing equal shock (wave reflection) inter-
action. Errors for output wave strengths, wave width and wave
position. Comparison of under resolved simulation and prediction. 39
2.10 Case 3. The contact-shock interaction (step down). Errors for
output wave strengths, wave width and wave position. Compari-
son of under resolved simulation and prediction. . . . . . . . . . 40
3.1 Comparison of the exponents from the approximate and the exact
similarity solutions for an inward propagating spherical shock wave. 44
3.2 The SNRP shock contact interaction. Expansion coefficients for
output wave strengths, wave strength errors, wave width errors
and wave position errors (linear model) for the initial shock con-
tact interaction. Here the base case input contact wave width
is zero. The final columns refer to difference between the linear
model (2.2) and the exact quantity. The errors in rows 4-12 refer
to the difference between the numerical solution on 100 cells and
the exact solution using 2000 cells. . . . . . . . . . . . . . . . . 48
3.3 The SNRP defined by the shock reflection at the origin. Expan-
sion coefficients for output wave strengths, wave strength errors,
wave width errors and wave position errors (linear model) for in-
put variation ±10%. . . . . . . . . . . . . . . . . . . . . . . . . 51
xv
3.4 The SNRP contact reshock interaction. Expansion coefficients for
output wave strengths, wave strength errors, wave width errors
and wave position errors (linear model). . . . . . . . . . . . . . 53
3.5 Predicted and simulated errors for output wave strengths, wave
widths and wave positions, output to interaction 3. The inward
rarefaction and contact strengths are expressed dimensionlessly as
Atwood numbers. The outward shock strengths are in the units of
Mach number. The width and position errors are in mesh units.
The wave strength errors are expressed as mean ± 2σ where σ is
the ensemble STD of the error/uncertainty. . . . . . . . . . . . 55
3.6 The contribution of each interaction to the mean value of the total
error in each of three output waves at the output to interaction 3,
for 100 and 500 mesh units. Units are dimensionless and represent
the error expressed as a fraction of the total wave strength. The
last two rows compare the total of the mean error as given by the
model to the directly observed mean error. The columns I.R., C.,
and O. S. are labeled as in Fig. 3.1, Right frame. . . . . . . . . . 57
6.1 Case 4. The SNRP defined by the crossing of two rarefactions.
Expansion coefficients for output wave strengths (linear model)
for input variation ±10%. . . . . . . . . . . . . . . . . . . . . . 72
6.2 Case 5. The SNRP defined by the contact rarefaction interaction.
Expansion coefficients for output wave strengths (linear model)
for input variation ±10%. . . . . . . . . . . . . . . . . . . . . . 74
xvi
6.3 Case 6. The SNRP defined by the shock shock overtake (two
waves of the same family). Expansion coefficients for output wave
strengths (linear model) for input variation ±10%. . . . . . . . . 77
6.4 Case 7. The SNRP defined by the crossing of two compressions.
Expansion coefficients for output wave strengths (linear model)
for input variation ±10%. . . . . . . . . . . . . . . . . . . . . . 78
6.5 Case 8. The SNRP defined by the contact compression inter-
action. Expansion coefficients for output wave strengths (linear
model) for input varation ±10%. . . . . . . . . . . . . . . . . . 79
6.6 Case 9. The SNRP defined by the crossing of two rarefactions.
Expansion coefficients for output wave strengths (linear model)
for input variation ±10%. . . . . . . . . . . . . . . . . . . . . . 80
6.7 Case 4. The crossing of two rarefactions. Predicted and simulated
errors for output wave strengths, wave widths and wave positions. 82
6.8 Case 5. The contact rarefaction interaction. Predicted and sim-
ulated errors for output wave strengths, wave widths and wave
positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.9 Case 6. The shock shock overtake. Predicted and simulated errors
for output wave strengths, wave widths and wave positions. . . . 84
6.10 Case 7. The crossing of two compressions. Predicted and sim-
ulated errors for output wave strengths, wave widths and wave
positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
xvii
6.11 Case 8. The contact compression interaction. Predicted and sim-
ulated errors for output wave strengths, wave widths and wave
positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.12 Case 9. The crossing of two rarefactions. Predicted and simulated
errors for output wave strengths, wave widths and wave positions. 85
xviii
Acknowledgements
I would like to express my profound gratitude to my advisor, Professor
James Glimm, for suggesting this important and exciting thesis topic and for
his advice, support and guidance toward my Ph. D. degree. He taught me not
only the way to do scientific research, but also the way to become a professional
scientist. He is my advisor and a lifetime role model for me.
I am also deeply indebted to the support of Professor Xiaolin Li. His
scientific vigor and dedication makes him a great mentor and a good friend.
I would like to thank Drs. John Grove, Kenny Ye, Zhiliang Xu, Roman
Samulyak, Dahai Yu, Yongming Zhang and Ning Zhao, from whom I have
learned many important scientific and mathematical skills. Work has been
more productive and life a little easier thanks to them.
I would also like to thank all my friends for their friendship and encour-
agement during my four year study as a graduate student at Stony Brook.
It has been great to have so many friends who can share my sense of hu-
mor, only a few yards away. In particular, I would like to mention Taewon
Lee, Ming Zhao, Erwin George, Andrea Marchese, Xinfeng Liu, Yoonha Lee,
Srabasti Dutta and Xiaofei Fan. They have shared with me many interesting
and inspiring ideas.
Throughout my academic career, the constant support of my parents and
my husband has always motivated me to strive forward. Their unconditional
love has never been affected by the physical distance between us. My disser-
tation is dedicated to them.
During my years here at Stony Brook I have grown and matured both
personally and professionally. I have had my greatest life experience, as well
as the worst. I am grateful to have gone through it all, and I look forward to
what comes next. I thank all who have helped me through this leg of journey.
xx
Chapter 1
Introduction
We begin Chapter. 1 with a background review of uncertainty quantifi-
cation to numerical simulations and the goal of our research study. In Sec. 1.2
and Sec. 1.3, we introduce the FronT ier Front Tracking algorithm for numer-
ical simulation and the wave filter as the diagnostic tool.
1.1 Background
Our approach to numerical solution errors is motivated by needs of un-
certainty quantification. Specifically the Bayesian likelihood is (up to normal-
ization) a probability, which specifies probability of occurrence of an error of
any given size. Unlike other authors [2, 6, 26, 33] who usually use observational
errors or expert opinions to form the probability model for the likelihood, our
approach is to use solution error models for the likelihood. We provide a
scientific basis for the probabilities associated with numerical solution errors.
The authors are not aware of comparable error analysis studies, but of course
numerical simulation errors have been long studied from different points of
1
view.
An early focus of numerical error modeling was round off errors. For the
hyperbolic systems we study, modern 64-bit processors with double precision
arithmetic appear, as a practical matter, not to be sensitive to this class of
errors, while they are difficult to analyze theoretically. A more common ap-
proach to error analysis in numerical analysis is the study of the asymptotic
behavior of errors under mesh refinement. The method of asymptotic analysis
of numerical solution errors is so old and well established that it is difficult to
cite its origins [38]. This is a useful approach, and one we refer to in the case
of well resolved simulations. However, we want an analysis which is also appli-
cable to the pre-asymptotic case of under resolved simulations as these are so
typical of practical studies of realistic complex physical systems. Moreover, the
coefficients which multiply powers of ∆x in the asymptotic expressions cannot
be determined theoretically. A third main theme in the analysis of errors is
the use of a posteriori error estimators. The method of a posteriori analysis
aims to construct an upper bound on the solution error, either theoretically
validated or based on numerical experiments [1, 4, 36, 37]. This method has
been difficult to apply to nonlinear hyperbolic systems, and in any case does
not answer the questions addressed here. We seek to characterize the error,
not just to bound it. Moreover, a posteriori methods are most fully developed
and justified theoretically for elliptic problems, and have only a partial or pre-
liminary development for the shock interaction problems we consider here. A
fourth approach to errors is to regard them as due to input uncertainties. In
this point of view, uncertainty analysis is a mapping of input random variables
2
to output random variables.
We are primarily interested in errors in a preasymptotic range, so while
these methods provide a theoretical framework which is valuable to our anal-
ysis, they do not provide the detailed estimates which we require. In any case
our use of wave filters to diagnosis error leads to more precise measures of
error than is normally considered in the asymptotic analysis.
In contrast to the first three of the above methods, we analyze the errors
statistically. The statistical models are simple, and in this sense, we analyze
only the central portion of the error statistics, not their tails. We use a Gaus-
sian error model, and thus we identify the mean error and the variance. The
mean error is a systematic error, and it can in principle be used to modify or
“post process” and error-correct the approximate solution. The variance is a
measure of the error variability. One may question the idea that numerical
errors can be modeled statistically or that the errors are variable, when each
simulation is totally deterministic. This philosophical question has an easy
answer: determinism lies in the eye of the beholder. In other words, the mod-
eling of a natural phenomena (tossing a coin, for example) as a probabilistic
or deterministic event depends on the level of detail included or excluded from
the model. Thus we argue only why it is convenient or nearly essential to
omit from the modeling of error information needed to make the error analysis
deterministic. The variable (as opposed to the systematic) part of the error
depends on “accidental” features of the numerical model, such as the sub grid
location of waves and wave interaction points relative to the mesh cell edges.
Clearly a deterministic model requiring such data would be too cumbersome
3
for use in practice, and thus a probabilistic model is preferable. Indeed, the
essentially probabilistic aspect of round off error is well recognized. Given that
the complexity of a statistical model is needed, we found that simple linear
error models were sufficient for our analysis [11]. The simple reason for this
pleasant turn of events is that the error is similar to a perturbation, and nor-
mally a small perturbation. Thus the error of a strongly nonlinear problem
still has a useful linear expression, at least as far as our analysis of the error
has progressed.
In contrast to the fourth approach to uncertainty quantification above as
a mapping of uncertainty from input to output, we allow for errors generated
within the solution processes. Thus we subsume and expand on this point of
view.
Simulation errors typically consist of
• position errors in the location of the travelling wave discontinuities or
sharp solution gradients;
• wave width errors in the numerical vs. the physical width of the travelling
waves;
• solution state errors in the smooth regions bordering the regions with
discontinuities or sharp solution gradients.
Any or all of these errors may arise in the input data to the Riemann problem.
Output errors, however, have two sources. Those arising from inaccuracies
in the solution algorithm are called created errors, while those that can be
4
ascribed directly to input error or uncertainty are called transmitted errors or
transmitted uncertainty.
In recent years, three major ideas have been used to develop an approach
to uncertainty quantification to numerical simulations:
1. A combined approach for forward as well as inverse propagation of un-
certainty [22, 23]. This combined approach is important when the use of
disparate sources of data, including data pertaining to observations of
full system performance, is important.
2. Error models for numerical solution errors [7, 17–19]. For multiscale
problems and complex multiphysics problems, under resolved simula-
tions and accompanying simulation error are frequently unavoidable in
practice.
3. Parameterizations, comprehensibility, and validation of simulation error
models [16]. This step allows testing and validation to occur in somewhat
idealized situations, less complex than the full system simulations, but
still applicable to them.
Our principal concern in this dissertation is to develop a method for
determining solution errors in shock wave interaction problems. Our strategy
takes advantage of the fact that shock problems typically consist of smooth
regions which separated by discontinuities (actually narrow regions with strong
solution gradients). The method consists of three main steps.
5
1. Determine solution error models for a comprehensive set of elementary
wave interactions. These are summarized as a set of input/output rela-
tions for the errors in such interactions;
2. Construct wave filters that decompose a complex flow into approximately
independent components consisting of elementary waves;
3. Formulate a composition law that constructs the total solution error at
any space-time point in terms of errors from repeated elementary inter-
actions.
We validate this method by predicting the errors in the composite (com-
plex) simulation with the errors calculated using the composition law [11]. The
results are derived by a study on errors in the solution of Riemann problems.
The Riemann problem is a simple jump discontinuity between two hydro-
dynamic states in one spatial dimension; its idealized solution is called the
Riemann solution [3]. We study the statistical numerical Riemann problem
(SNRP). In SNRP, incoming shock waves have finite width determined from
numerical as well as physical considerations, if specified numerically. Incoming
rarefaction and compression waves have physical time dependent widths. Also
we note that the numerical algorithm that solves the Riemann problem cre-
ates (as well as propagates) errors. We will show that, to fairly good accuracy,
one can model the errors in the outgoing waves as affine linear, i.e., constant
plus linear (or perhaps bilinear) statistical expressions in the strength of the
incoming waves.
6
1.2 FronT ier Front Tracking Algorithm
Since most of our work is based on numerical simulations using FronT ier
code, we present here a brief introduction to this code and its Front Tracking
algorithm.
FronT ier code is based on front tracking, a numerical method which
regards the fluid interface as an evolving, lower dimensional mesh. The Front
Tracking method initiated by Richtmyer and Morton [39] is designed to pro-
vide piecewise smooth solutions as well as distinguished discontinuities which
separate the smooth solutions in solving the system of hyperbolic partial dif-
ferential equations. Fig. 1.1 is a flow chart taken from [14] describing the front
tracking computation in FronT ier. For more details, see [13–15].
Two tasks need to be accomplished at one time step in the Front Track-
ing method. The first is to dynamically evolve the front. The second is to
calculate the numerical solutions in smooth regions surrounded by the fronts.
Corresponding to these two tasks, the Front Tracking method consists of the
following modules.
1.2.1 Interface propagation
An interface is a collection of geometric objects, such as POINT s, CURV Es,
and SURFACEs, that correspond to zero, one, and two dimensional mani-
folds respectively. The interfaces are represented explicitly as lower dimen-
sional meshes. They divide the computational domain into connected regions.
1. The interface points are first propagated normally. By computing the
7
Figure 1.1: Flow chart for the front tracking computation. With the exceptionof the i/o and the sweep and communication of interior points, all solutionsteps indicated here are specific to the front tracking algorithm itself.
8
solution to the local Riemann problem with initial states being those on
either side of the interface point, a wave speed and hence a new position
for the interface point are predicted. These values are only intermediate,
as they do not account for other wave interactions and they assume
constant states on both sides of the discontinuity. The Riemann solution
is usually solved by Newton or secant iteration methods.
2. The method of characteristics is used to linearly trace back to states at
the previous time step. For tracked contact discontinuities (where den-
sity and energy are discontinuous, but pressure and normal velocity are
continuous) as we study in this document, there is one characteristic on
either side of the discontinuity. The states at the previous time step (at
the foot of each characteristic) are computed by interpolating between
the original states on either side of the interface, and those one mesh
unit away in the normal direction.
3. These interpolated states with the Rankine-Hugoniot conditions deter-
mine a solution to the discretized characteristic equations, which pro-
vides an update to the left and right states of the predicted new position.
4. With these updated states at new position, we solve a Riemann problem,
and compute an updated wave speed as well as even further updated
states.
5. The front speeds from both Riemann problems (step 1 and 4) are aver-
aged to compute the final front speed, and this speed is used to compute
the final new position of the interface point.
9
6. The interface states are then updated by a tangential sweep, which uses
a chosen interior solver with a stencil centered at the new interface point.
7. Any tangles in the interface are resolved.
8. At user-defined intervals, the interface points are redistributed to reduce
the variance in size and aspect ration of the segments (2D) or triangles
(3D) making up the interface.
1.2.2 Interior computations
A connected region in the domain separated by the interface is repre-
sented by a component. Therefore, each grid node is associated with a specific
component in addition to the state variables. The interior states are updated
by finite difference schemes.
The interior computations are dimensionally split, so that we may use
simpler, one-dimensional schemes. The order of the one-dimensional sweeps
is changed periodically to avoid incurring first order simulation error (for de-
tails, see [35]). Computations near the fluid interface use ghost cells [21] to
avoid crossing the interface, keeping the different fluid computations entirely
separate. Computational experiments have shown the use of ghost cells to be
reasonably conservative [14]. Our choice of interior solver for the work in this
document is the MUSCL scheme.
10
1.3 The Wave Filter
We introduce diagnostic windows that measure the solution state in one
of the constant regions between the waves as well as wave filters that diagnose
the wave type (only regions with a single wave pass through the filter). In doing
so, we first summarize and then extend the results of [32]. The moving window
in the wave filter has an initial width of 5 cells for shock and rarefaction waves
and 11 cells for contacts. The choice of these parameters appears to be suitable
for most higher order Godunov schemes. In this window, a Riemann problem is
solved using the extreme left and right states as input. The Riemann solution
has 3 outgoing waves, whose strengths are assessed dimensionlessly in terms
of density and pressure differences and ratios. According to these strengths,
and a suitable cutoff for the strength, we identify from zero to three of the
waves as strong, and only the case of a single strong wave is analyzed further.
If adjacent or overlapping windows show a single identical wave, the windows
are merged, so that the full width of the wave will be brought into a single
window. This merging of adjacent or overlapping windows of increasing size
continues recursively until the same wave type fails to show up in the adjacent
windows.
Wave profiles are reconstructed using fitting functions of the form:
ρ(x, t) = ρ− +ρ− − ρ+
2
(f(
x − xc(t)√2σ
) + 1
)(1.1)
where ρ± refer to the asymptotic values for density ahead or behind the wave,
xc(t) is the moving center of the wave, and 2σ is the measure of the wave
11
width. The fitting function f(x) is either the erf function
f(x) = erf(x) =2√π
∫ x
0
e−t2dt (1.2)
for contact or shock waves, or a linear ramp:
f(x) =
−1
x
1
x < −1
−1 < x < 1
1 < x
(1.3)
for rarefactions and compressions.
States identified as within an active region for a single shock or contact
wave by the filter are fit to an error function. The fitting allows determination
of the location of the wave (with subgrid accuracy, up to O(∆x2)), and its
width. See Fig. 1.2, Left. For single rarefaction or compression waves, the
waves are fit to a straight line segment, linear in the characteristic speed
variable. Thus the region of constant and variable states are fit to three
straight line segments, the two extreme ones being constants. See Fig. 1.2,
Right. The width for a shock or contact wave is defined in terms of the
error function fit to the shock or contact wave profile. Let σ be the standard
deviation that enters into the definition of the error function. Then the width
(in units of ∆x) is the distance needed for a 2σ transition (between 2.3% and
97.7%) of the jump in the density (for a contact wave) or in the pressure (for a
shock wave). The width of a rarefaction or compression wave is defined as the
distance between the edges of the central linear piece for its piecewise linear
12
description. The position is defined, with subgrid accuracy, as the position of
the mean value, at a point half way through the jump.
Figure 1.2: Schematic diagram illustrating the operation of a wave filter. Left:computational data (squares) are fit to an error function. The error functiondepends on four parameters, a position, a width, and two asymptotic values.These determine the wave position, width and height, with subgrid accuracy.Right: a piecewise linear construction is fit to the rarefaction or compressionwave data.
The wave filter is the fundamental diagnostic tool that identifies individ-
ual waves, here within the solution of a numerical Riemann problem and in
Sec. 2.3 within the solution of a complex wave interaction problem. We note
immediately a limitation of the methodology, at least as presently developed.
The definition of the wave filters assumes that the individual waves in the
Riemann problem have separated. For sufficiently coarse grids in the wave in-
teraction problem of Sec. 2.3, the waves will enter into new interactions before
clearly separating as they leave an earlier interaction. A second, and related
limitation concerns the relaxation of the left and right states at the edge of a
13
wave to their far field values, an issue studied in Sec. 2.2.1. If a subsequent
interaction occurs before this relaxation is complete, the associated errors will
be “frozen” into the input and output of this later interaction. We will assess
this issue in Sec. 2.3.
1.4 Dissertation Organization
In Chapter. 1, we gave a background review of uncertainty quantification
to numerical simulations and the goal of our research study. We have also
introduced the FronT ier Front Tracking algorithm for numerical simulation
and the wave filter as the diagnostic tool. The rest of this dissertation will be
devoted to analyze the errors in numerical solutions of shock physics problems.
Chapter. 2 present the composition law for complex 1D shock wave in-
teraction problem in planar geometry. The problem is generated by a shock
wave interacting with a contact in the vicinity of a wall. Multiple reflections
between the wall and contact generate a large number of Riemann problems,
that comprise the major features of this problem. The errors can be com-
puted in two ways, directly by statistical analysis of the data and combining
the errors created by and propagated through the individual Riemann prob-
lems using the composition law. The comparison of errors for the complex,
multi-interaction problem, thus determined in two ways, provides validation
for our proposed composition law for errors.
In Chapter. 3, we study the composition law in spherical geometry. We
explore complications introduced by spherical flow in the analysis of errors in
the numerical solutions. We also conduct a detailed analysis of the errors to
14
understand the contribution of each wave interaction to the errors observed at
the end of the simulation.
The conclusion and a discussion of remaining open problems are presented
in Chapter. 4 and 5. Finally, in the appendix, we present the complete set of
Riemann problems (planar geometry) that are studied.
15
Chapter 2
Error Analysis for Planar Geometry
2.1 Problem Setup
We consider, in one spatial dimension, the interaction of a shock wave
with a contact located near a reflecting wall. The base case for the first wave
interaction is defined in Table 2.1. In this table, P represents the pressure
ratio for the shock wave, defined as P = (P2 −P1)/(P2 +P1); A represents the
Atwood number for the contact, A = (ρ2 − ρ1)/(ρ2 + ρ1). P and A are used
as the wave strength for the shock and the contact.
Shock Contactvariable Left Right Left Right
ρ 3.973980 1.0 1.0 10.0p 1.337250 0.001 0.001 0.001v 1.0 0.0 0.0 0.0γ 1.67 1.67 1.67 1.67
P = 0.999 A = 0.82
Table 2.1: Parameters that define the base case for the shock contact interac-tion.
16
This base case coincides with the base case for the shock contact inter-
action to be studied in Sec. 2.2. We further specify the wall location as 1.5
units to the right of the initial contact location. The transmitted shock, after
interaction with the contact, progresses to interact with (i.e. reflect off) the
wall. This interaction will also be studied in Sec. 2.2. Subsequently, there are
a number of reverberations, of reflected rarefactions and compression waves,
between the contact and the wall and between a new contact formed by a shock
overtake interaction and the original contact. The new contact is clearly visi-
ble in Fig. 2.1 (left), as the vertical line near the left border, starting at a time
about t = 5.2. The interactions are illustrated by the space time contour plots
of the density, shown in Fig. 2.1, left and pressure contours, right. In Fig. 2.2
(left), we show the type and location of the waves, as determined by our wave
filter analysis program. Both figures refer to the base case. The build up of
complex wave patterns is evident. Ten Riemann problems are extracted from
the complex wave interaction problem. A schematic representation of these
ten Riemann problems is given in Fig. 2.2 (right).
The ensemble of 200 initial conditions is defined by a Latin hypercube
variation shock and contact strength by ±10% about the base case defined as
above.
2.2 The Statistical Numerical Riemann Problem
The SNRP introduces errors (modelled as random) in addition to prop-
agating errors or uncertainty from input to output. The waves in the SNRP
have a finite width and the solution algorithm in the SNRP has only finite
17
Figure 2.1: Left. Space time density contour plots for the multiple waveinteraction problem studied in this section. Right. Pressure contour plots forthe base case considered here.
accuracy. Because of the possible finite width of the input waves, the problem
and its solution are not strictly scale invariant, and so we consider a general-
ization of the Riemann problem.
A statistical distribution of numerical incoming waves and starting states
determines the SNRP. Its solution gives the output waves, each of which gen-
erates the same type of data Thus we define the SNRP as a statistical (non-
deterministic) mapping from a statistical input wave description to a statistical
output wave description.
The statistics of the SNRP mapping function arise from grid errors, and
from the random placement of a travelling wave relative to the centers of the
finite difference lattice. Our objective in this section is to build up a library
18
Figure 2.2: Left: Type and location of waves as determined by our wave filteranalysis for the base case considered here. Right: Schematic representationof the waves and the interactions, with labels for the interactions, taken fromthe left frame.
of statistical input-output relations that will include all Riemann problems to
be encountered in Sec. 2.3. This library will be used to predict results for the
multi-wave error and uncertainty analysis based on a multi-path scattering
formula.
2.2.1 The Isolated Shock and Contact Wave
We start with the analysis of the ensemble averaged mean width of a
single (non-interacting) wave.
Fig. 2.3 (left) shows the expected narrow and time independent (∼ 2∆x)
shock width. Among the several factors contributing to wave strength and
19
speed errors, we mention the finite accuracy of the Riemann solution root
solver used in the numerical scheme, and the numerical (finite difference) na-
ture of the solution. The latter arises in two ways, the relaxation to a constant
ambient state and the finite rate of convergence under mesh refinement, both
applicable on the post shock or up stream side of the shock wave.
Figure 2.3: Ensemble mean shock width and the standard deviation of theshock width (left frame). The mean width, equal to about 2∆x, is much largerthan the standard deviation, indicating that the mean width is essentially adeterministic feature of the solution. Convergence properties of the travellingwave to the steady state values on each side of the wave (right frame). Thestraight line in the right frame is the asymptote to the exponential convergencerate, with slope 0.01 in units of ∆x.
According to the theory of travelling waves for the viscous Riemann prob-
lem [40], considered as a model for numerically generated travelling waves, we
expect an exponential approach of the numerical shock profile to its limiting
values at x = ±∞. The error occurs on the upwind side of the shock, while
20
the down wind states converge identically to their far field values within a few
mesh blocks. See Fig. 2.3 (right). We measure the local extrema in the error
dimensionlessly as emax = |(ρ − ρ∞)/ρ∞| where ρ∞ is the far field density.
Then we model emax(n∆x) = c exp(−λn) where n is the distance from the
shock front in mesh units. We find λ = 1.0 × 10−2 and c = 2.4 × 10−4 for
the shock wave defined in the base case, see Table 2.1. The first extrema is a
local maximum, occurring about 4∆x from the center of the shock front. The
details of the shock error behavior will be sensitive to the numerical method,
but the general form of the error, as it is derived from a mathematical theory,
should be somewhat universal.
Fig. 2.4 (left) shows the larger contact width wc ∼ cct1/3 growing from 2
to 30 cells with a rate asymptotically proportional to t1/3. Similar asymptotics
have been observed by Harten [25] for an ENO scheme. The rate t1/3 results
from the second order accuracy of the method used here. The numerical
diffusion is sensitive to the direction of mixing. For the flow from high density
to low, the numerical mixing is of heavy fluid into light. We call this the step
down problem. For the step down problem, shown in Fig. 2.4 (left), we find
cc ∼ 1,
The reverse, called the step up problem, flows from light to heavy fluid. It
mixes small amounts of light fluid into heavy, an effect less noticeable in terms
of the diffusion width, especially for large density contrasts. For a step up
flow, we find wc ∼ min5, cct1/3. See Fig. 2.4. As a partial explanation of this
difference between the step down and the step up problems, we note that the
spreading is primarily associated with the up stream side of the contact, and
21
Figure 2.4: Ensemble mean contact width for isolated noninteracting waves.Because the width is entirely grid related, we record width in units of ∆x andtime in units of the number of time steps. The standard deviations are alsoplotted, and are the points to the extreme left in each frame. Left (step down):we observe an increase from 2 cells to 30 over 104 steps and an asymptoticgrowth rate cct
1/3, where cc ∼ 1 depends on the flow Mach number. Thestraight line in the left frame is the asymptote to the contact width, withslope 3. Right (step up): We observe a bound on the contact width.
that continued spreading (the t1/3 asymptotics) depends on the up stream flow
being subsonic. The higher sound speed in the light fluid gives a supersonic
upstream state for the step up problem but not for the step down problem, for
the flow parameters considered here. These properties appear to be sensitive
to the details of the numerical algorithm, and specifically to the form of the
limiter employed. We have used a MUSCL algorithm here.
For some aspects of the solution error, the probabilistic error formalism
is more general than is required. When the standard deviation of the error
is much smaller than the mean error (when the coefficient of variation, their
22
ratio, is close to zero), then the error is essentially deterministic, and the
probabilistic formulation is unnecessary. This is the case in Fig. 2.4, with the
standard deviation of the width, shown to the left scale of Fig. 2.4, significantly
smaller than the mean width.
2.2.2 Shock Contact Interactions
We study wave strength, width and position errors after the numerical
shock-contact interaction, the wave interaction of a shock wave moving (to the
right) into a contact (density increase, or step up case). This wave interaction
initiates the complex series of interactions to be studied in Sec. 2.3. The base
case for this interaction is also defined in Sec. 2.1, see Table 2.1. The shock
strength is given with a pressure ratio 1337 (P = 0.999), and the contact
strenth is given with a density ratio 10 (A = 0.82). The equation of state
is a gamma law gas, with γ = 1.67. In Fig. 2.5 (left), we show the time
development of the widths of the two incoming waves and the three outgoing
waves for the shock-contact (step up) interaction. This interaction occurs early
in the development of the contact. In Fig. 2.5 (right), we show the position
errors as a function of time. We adjust the ensemble to have a common time
and location of interaction.
We represent the wave properties as a quadruple
wak = (ωa
k , λak, s
ak, p
ak) , (2.1)
where ω is a wave strength, λ is a wave width, s is a wave speed error, and
23
Figure 2.5: Left: ensemble mean shock and contact widths before and after ashock contact (step up) interaction and standard deviations. Right: ensemblemean shock and contact position errors as a function of time, expressed in gridunits. Step up case.
p is a position error. Also a = i for input wave and a = o for output wave.
We choose dimensionless variables to measure the wave strengths: a modified
Atwood number A = (ρ2 − ρ1)/(ρ02 + ρ0
1) to measure the contact strengths,
and a similar expression built out of the pressures, P = (P2 − P1)/(P02 + P 0
1 ),
for the other wave types. Here the quantities ρ0i and P 0
i denote densities
and pressures from the base case associated with the ensemble (we specify
a starting state for the incoming waves and a base case for the strong wave
strengths). We consider variation about this base case by ±10% in the density
ratio (for contacts) or pressure ratio (for all other waves). The wave widths
are measured in units of mesh spacing. The wave position errors are specified
24
in mesh units, after an isolated transient period. Within this formulation, we
can describe the output wave errors by an expression linear in the two input
wave strengths, i.e. linear in the product of the input wave strengths.
For input wave strengths wi1, wi
2 (i ≡ in) and output wave strengths wo1,
wo2 and wo
3 (o ≡ out) (ordered from left to right), the multinomial expansion for
the output is defined by its coefficients αk,J , for J a multi index, J = (j1, j2).
The expansion has the form
wok =
∑J
αk,Jwi,J , (2.2)
where wi,J = (wi1)
j1(wi2)
j2 . The coefficients αk,J depend parametrically on
the base case Riemann problem, about which a specified variation is allowed.
Given a statistical ensemble of input and output values wi and wo, we use a
least squares algorithm to determine the best fitting model parameters αk,J ,
for any given polynomial order of model. We use (2.2) variationally, that
is to map input variation (about the base case for the ensemble) to output
variability. In other words, (2.2), which is a formula for wave strengths, implies
a similar formula with different but computable coefficients αk,J , in which
all ω’s are defined as variations from the base case, so that they represent
uncertainty or error. We consider the case of a linear input-output relation,
ωok = αk,0 +
∑j αk,jω
i,j. We have an expansion similar to (2.2) for wave widths
and wave position errors.
We begin with the analysis of the SNRP at the ensemble averaged level.
We present the mean model analysis in Table 2.2, with ±10% variation about
25
the base case. The input contact width has been set to zero, as part of the
specification of this SNRP.
variable \ coef const ωi1 ωi
2 error(r. sonic) (contact) L∞ STD
ωo1 (l. sonic) -0.208 0.454 0.251 0.47% 0.001
ωo2 (contact) -0.042 0.000 0.912 0.03% 0.0001
ωo3 (r. sonic) -0.286 1.004 0.346 0.30% 0.001
λo1 (l. sonic) 2.184 -0.563 0.000 122% 0.240
λo2 (contact) 4.725 0.110 -1.466 0.67% 0.010
λo3 (r. sonic) 2.197 0.068 0.106 5.35% 0.057
po1 (l. sonic) 0.221 -0.014 0.023 27.1% 0.022
po2 (contact) 0.426 0.001 -0.092 1.78% 0.002
po3 (r. sonic) 0.332 -0.004 -0.099 3.47% 0.005
Table 2.2: The SNRP shock contact (step up) interaction. Expansion coeffi-cients for output wave strengths, wave widths and wave position errors (linearmodel) for input variation ±10%. Here the base case input contact wave widthis zero.
To read Table 2.2, we note that the first (wo1) row (labelled in the table
as wo1 (l.sonic)) lists coefficients α1,J for J = (0, 0), J = (1, 0), etc. These
coefficients are determined by a least squares algorithm, that minimizes the
expected, or mean error over the ensemble, in comparing the linear predictions
to the exact solution of the Riemann problem. The mean is taken over the
statistical ensemble. The last two columns describe errors in the model (2.2).
The column labelled L∞ is the maximum of the absolute value, over the en-
semble, of the relative error, expressed in per cent. It is thus a pointwise error
estimate. The maximum is computed using a sample size of 200, and it may
be sensitive to the choice of ensemble. The relative error is defined as (pre-
dicted - exact)/exact where exact is the result of the (exact) Riemann solution
26
and predicted is the value given by the finite polynomial (linear) model. The
column STD is the standard deviation of (predicted - exact). From the small
values of these errors for the linear model, as seen in Table 2.2, we conclude
that the linear model is adequate for many purposes.
The three variable (λ) rows in Table 2.2 represent wave width errors.
The large sup norm error for the width of the reflected shock results from
a few outliers, mostly but not entirely due to smaller shock widths than the
mean. The standard deviation for this quantity is about 10% of the mean
value, indicating that the error model is (on the whole) satisfactory, and that
the shock wave widths are not (mostly) fluctuating greatly. The outliers are
mainly associated with time steps and realizations for which the (narrow)
reflected shock has at most one internal mesh points. For these cases, our
filter tool for assessing the numerical shock width and position is not effective,
so the outliers can be viewed as a breakdown of the diagnosis methodology.
We also study the wave position errors. Fig. 2.5 (right) shows the position
errors as a function of time. Following the initial transient, the position error
is constant, reflecting convergence of the wave speed to its exact value. The
errors in the wave position rows of Table 2.2 present this constant error, given
in mesh units. All position errors are subgrid. The standard deviations are
smaller than the means, indicating that the errors are basically deterministic.
The L∞ position error is the supremum of the relative error. It is an error
in the model of the error, i.e. the error in the error. Occasional ensemble
members with very small (exact) position error produce a small denominator
in the relative error, (model error)/(exact error). Thus the large entries in
27
this column (also in other tables) do not represent a deficiency in the error
model. We see similar and more extreme L∞ per cent errors in later tables,
with the same cause. Note that the standard deviation is comparable to the
mean position error, so that occasional instances of nearly zero error are to be
expected.
2.2.3 Shock Crossing Shock Interactions
Here we study the reflection of the shock off the wall, a special case of
the shock crossing shock interaction. We can ignore the shock wave width
parameters, as these are narrow and deterministic. We only consider one
output wave position error, po1. See Table 2.3. The shock wave strength errors
(the far field, large separation errors) are small.
variable \ coef const ωi1 error
(r. sonic) L∞ STDωo
1 (l. sonic) -0.002 0.716 0.014% 0.000032λo
1 (l. sonic) 2.291 -0.422 7.923% 0.062po
1 (l. sonic) 0.060 -0.039 5065% 0.009ωo
2 (contact) 0.057 0.0003 24.4% 0.005λo
2 (contact) 5.9 50% 0.7
Table 2.3: The SNRP defined by the crossing of two shocks. Expansion coef-ficients for output wave strengths, widths and position errors (linear model)for input variation ±10%.
The contact mode contributes an error, well known as shock wall heating.
The exact solution has no contribution in this mode for a wall reflection.
The error is the imprint on the entropy, temperature and density variables of
28
entropy errors made during the shock interaction process, apparently due to
shock oscillations. Since entropy can only increase, according to the second law
of thermodynamics, these errors do not cancel. Because the solution algorithm
conserves mass locally, we expect the spatial integral of the density errors
to cancel approximately. Since the velocity of the fluid at the wall is zero,
these errors do not move, and remain permanently attached to the wall. As
a numerical wave, the error is a standing wave. We do not have a theoretical
model for the form of these errors. Due to this lack, our fitting of the errors
will be less precise than those discussed elsewhere in this paper. We define the
wall error width to be the distance to the wall in mesh units of the furthest
location for which the density error is at least twice the background noise in
the post shock region, or about 1% of the base case density. This width is
about 6∆x. Also the wave strength error is defined dimensionlessly as the L1
error in density, divided by ρ0∆x, where ρ0 is the base case post shock density
after the wall reflection.
2.2.4 The Contact Reshock Interactions
After reflection from the wall, the transmitted lead shock wave re-crosses
the deflected contact. This is a step down interaction. We have one input
and one output wave width parameter, both for the contact. According to the
analysis of Sec. 2.2.1, the contact width is is modelled as cct1/3 where both
the width and t are expressed in mesh units. This formula is accurate after
some 50 time steps, according to Fig. 2.5, and the Table 2.4 entry λo2 = cc in
this formula. We form a linear model for this constant in this expression in
29
Table 2.4. The rarefaction width has the form constant + rate × time. We
find very small errors in the rate, not tabulated here. The entry λo3 refers to
the constant, which gives an offset for the centering of the rarefaction wave.
This entry is expressed in mesh units.
variable \ coef const ωi1 ωi
2 error(contact) (l. sonic) L∞ STD
ωo1 (l. sonic) 0.282 -0.314 0.645 0.57% 0.0008
ωo2 (contact) 0.013 0.819 0.118 0.20% 0.0003
ωo3 (r. sonic) -0.128 0.143 0.468 0.41% 0.0004
λo1 (l. sonic) 2.383 0.754 -1.307 5.47% 0.038
λo2 (contact) 0.909 0.011 0.216 1.00% 0.005
λo3 (r. sonic) 3.619 0.151 -0.974 14.8% 0.138
po1 (l. sonic) 0.242 0.043 0.042 10.4% 0.014
po2 (contact) -0.036 0.045 0.066 75.5% 0.008
po3 (r. sonic) -0.447 0.078 -0.036 16.7% 0.029
Table 2.4: The SNRP shock contact (step down) interaction. Expansion co-efficients for output wave strengths, widths and position errors (linear model)for input variation ±10%.
2.3 The Composition Law
Here we add up all the pieces. We introduce a formula for combining the
wave interaction errors defined in Sec. 2.2 for isolated Riemann problems, to
yield the error for arbitrary points in the complex wave interaction problem.
The formula is validated for fully resolved simulations and it is shown to be
partially correct and partially incorrect for under resolved simulations.
30
2.3.1 A Multipath Integral for a Nonlinear Multiscat-
tering Problem
We begin with a formula expressing the error in a given Riemann problem
R0 as multinomial expansion associated with initial waves and errors located
inside its domain of dependence. For a 1D shock wave interaction problem,
think of the solution as being primarily composed of localized waves, interact-
ing through Riemann problems and generating outgoing waves, that further
interact in the same manner. The interaction of waves generates a planar (1D
space and time) graph, the vertices of which are the Riemann problems and
the bonds are the travelling waves, between Riemann problem interactions.
Starting from a given Riemann problem (vertex) or wave (bond), we can trace
backward and determine its domain of dependence. Call this graph G.
For each Riemann problem, we consider two types of vertices, correspond-
ing to the constant and linear terms in the parameterized approximate solution
and error terms developed in Sec. 2.2. We treat the linear terms separately,
as they allow a simple propagation law,
SL =
∫w(t = 0)dω (2.3)
where w(t = 0) is a vector representing the strength of the time zero wave and
its error or uncertainty, evaluated at the beginning of the path ω, and SL is the
purely linear propagation contribution to a final time error. The path space
integral dω is taken over all paths progressing in time order through G from
the initial time to the final vertex, with each term weighted by the appropriate
31
linear factors from the formula for the approximate solution of the Riemann
problems transversed. This path space representation makes evident the point
that the solution SL is that of a multiple (linear) scattering problem.
Figure 2.6: The solution and its errors at the point (x, t) can be obtainedby “adding up” the solution and errors for the waves within the domain ofdependence
The amplitude S at the final time (vertex of G) can similarly be thought
of as a solution of a nonlinear multiple scattering problem, leading to a repre-
sentation in terms of multipath integrals. To allow nonlinear (constant) inter-
actions, we re-introduce the vertices from these other terms. Let V = V(G) be
the set of vertices of G, and let B ⊂ V be a subset of V where constant terms
occur. The total amplitude S will then be a sum over terms SB indexed by B.
For each v ∈ B, let Iv be the the interaction coefficient, taken from a table of
Sec. 2.2. We write
S =∑
B⊂V(G)
SB =∑
B⊂V(G)
∫ ∏v∈B
IvdωB . (2.4)
32
The multipath propagator dωB is a product of the individual propagators ω
for each single path, as in (2.3). The summation in (2.4) can be understood
schematically as the sum over all events within the domain of dependence of
the evaluation point (x, t) at the vertex of G. See Fig. 2.6.
2.3.2 Evaluation of the Multipath Integral
From prior work [7], we know that the dominant errors in the composite
solution are located within the leading shock and contact waves of the problem.
A portion of these errors are simple resolution errors (created errors at the
interaction). Wave strength errors can be modelled as only transmitting from
the initial uncertainty. Wave width errors shows up in the created errors only.
As for wave position errors, transmitted errors from the previous interaction
and created errors at current interaction should be combined together as total
position errors.
We develop formulas for the propagation of the initial uncertainty to the
output uncertainty of interaction 3, in Fig. 2.2 (right). The initial uncertainty
is reflected in the wave strength variables, according to the definition of the
ensemble of initial conditions. The transmission of the mean values through a
linear model is standard and is not detailed here. The transformation of the
variance is our major concern. Let B(l) with matrix entries β(l)jk be the matrix
that gives the linear transformation of these variables due to interaction l. We
note that the matrix entries β(l)jk are defined by the ωi columns and ωo rows of
Table 2.2 (linear; l = 1), Table 2.3 (l = 2), and Table 2.4 (l = 3). The output
to interaction 3 has three components, and we compute the variance of each,
33
labelled j = 1, 2, 3. We have
Var ωo(3)j = (β
(3)j2 )2Var ω
i(3)2 = (β
(3)j2 )2Var ω
o(2)1
= (β(3)j2 )2(β
(2)11 )2Var ω
i(2)1 = (β
(3)j2 )2(β
(2)11 )2Var ω
o(1)3
= (β(3)j2 )2(β
(2)11 )2
2∑k=1
Var (β(1)3k )2ω
i(1)k (2.5)
We also need to calculate formulas giving the transmission of position
errors through the various Riemann problems. For interaction with a wall (e.g.
case 2), the formula is elementary. Assuming no error in the wall position, let
pi and po denote input and output position errors for a reflection off of a
stationary wall, where the output is due transmission of error, i.e. due only to
the input, as opposed to Sec. 2.2, where there is no input position error and
the output position error is created during the interaction. Then we have
po = pi vo
vi(2.6)
where vi and vo are the incoming and outgoing wave speeds for the waves
involved in the wall reflection. For the interaction of two incoming waves, the
result is slightly more complicated. Each of the two terms in the formulas
below is due to the input error in one of the input waves. That error can be
computed by (2.6) if we perform the analysis in the frame in which the other
wave is stationary. The result is
po1 =
pi1(v
o1 − vi
2) + pi2(v
i1 − vo
1)
vi1 − vi
2
(2.7)
34
po2 =
pi1(v
o2 − vi
2) + pi2(v
i1 − vo
2)
vi1 − vi
2
(2.8)
po3 =
pi1(v
o3 − vi
2) + pi2(v
i1 − vo
3)
vi1 − vi
2
(2.9)
where pij is the position error of the incoming contact wave (j = 1) or left facing
shock (j = 2). The complete position error model is obtained by adding the
results of (2.7) - (2.9) to those of Table 2.4 for the position errors created at
the interactions. For interactions 1 - 3, we model the wave width (error) as a
created error only.
2.4 Numerical Results
2.4.1 Errors in Fully Resolved Calculations
We regard a calculation as resolved if all (the principal) waves have sep-
arated, with converged left and right asymptotic states, before they interact
with another wave. For this type of simulation, we choose 500 mesh cells in
our basic simulation study. We examine errors in wave strength, wave position
and wave width, based on the graphical expansion given in Sec. 2.3.1, 2.3.2.
The wave strength errors are dominated by the transmission of error (or un-
certainty) from the initial conditions. In Tables 2.5, 2.6 and 2.7 we compare
the predicted error with the error computed directly, taken from a full solution
of the multiple wave interaction problem. The model for the prediction of the
error is satisfactory for all cases: the wave strength and its errors, the wave
width errors, and the wave position errors.
35
variable \ error Simulation Predictionmean wave strengths
ωo1 (l. sonic) 0.451 0.452
ωo2 (contact) 0.704 0.703
ωo3 (r. sonic) 0.999 0.998
wave strength errorsVar ωo
1 (l. sonic) 0.0008 0.0008Var ωo
2 (contact) 0.0019 0.0018Var ωo
3 (r. sonic) 0.0035 0.0036wave width errors
λo1 (l. sonic) 1.630 1.622
λo2 (contact) 3.636 3.635
λo3 (r. sonic) 2.346 2.352
wave position errorspo
1 (l. sonic) 0.220 0.226po
2 (contact) 0.313 0.312po
3 (r. sonic) 0.200 0.202
Table 2.5: Predicted and simulated errors for output wave strengths, wavewidths and wave positions, Interaction 1.
variable \ error Simulation Predictionmean wave strengths
ωo1 (l. sonic) 0.713 0.714
wave strength errorsVar ωo
1 (l. sonic) 0.0018 0.0018wave width errors
λo1 (l. sonic) 1.868 1.869
wave position errorspo
1 (l. sonic) -0.118 -0.092
Table 2.6: Predicted and simulated errors for output wave strengths, wavewidths and wave positions, Interaction 2.
36
variable \ error Simulation Predictionmean wave strengths
ωo1 (l. sonic) 0.520 0.519
ωo2 (contact) 0.674 0.674
ωo3 (r. sonic) 0.306 0.305
wave strength errorsVar ωo
1 (l. sonic) 0.0009 0.0010Var ωo
2 (contact) 0.0012 0.0013Var ωo
3 (r. sonic) 0.0004 0.0004wave width errors
λo1 (l. sonic) 2.097 1.982
λo2 (contact) 5.027 4.918
λo3 (r. sonic) 2.875 3.033
wave position errorspo
1 (l. sonic) -0.097 -0.105po
2 (contact) -0.003 0.013po
3 (r. sonic) -0.151 -0.134
Table 2.7: Predicted and simulated errors for output wave strengths, wavewidths and wave positions, Interaction 3.
37
2.4.2 Errors in Under Resolved Calculations
Here we allow 100 cells for the coarse grid simulation. This resolution
allows 10 cells between the contact and the reflection wall at the time of
interaction 3 and beyond. Since the contact has a width of 5 cells, since the
right facing rarefaction is about this size and since the wall has inaccurate
states in a region of several mesh blocks neighboring it, we are clearly at the
limit of the present diagnostic methods based upon the wave filter. For the
same reasons, the calculation is clearly under resolved. For this reason we are
not able to analyze data for the case of a coarse grid simulation with 50 cells
using the present version of our wave filter. Again we present the first three
interactions in detail, at 100 cell resolution, comparing the predicted to the
directly simulated errors. See Tables 2.8, 2.9, 2.10. We see good results for the
wave strengths and their errors and for the wave width errors, and poor results
for the comparison of position errors. This can be understood in terms of the
decay time for convergence to asymptotic large time values for the position
errors, an explanation that also accounts for the difference with the resolved
case, for which the simulated and predicted position errors agree. The position
errors have a relatively slower decay time. The other three quantities show a
high level of agreement between the resolved and under resolved cases. The
wave width error is expressed in grid units, and so should be the same in the
two cases differing in grid resolution only. For the wave strength entries, the
lack of dependence on grid resolution is due to the fact that these quantities are
dominated by the uncertainty expressed in the ensemble of initial conditions,
which is independent of grid resolution.
38
variable \ error Simulation Predictionmean wave strengths
ωo1 (l. sonic) 0.451 0.452
ωo2 (contact) 0.741 0.703
ωo3 (r. sonic) 0.996 0.998
wave strength errorsVar ωo
1 (l. sonic) 0.0008 0.0008Var ωo
2 (contact) 0.0022 0.0018Var ωo
3 (r. sonic) 0.0036 0.0036wave width errors
λo1 (l. sonic) 1.381 1.621
λo2 (contact) 3.498 3.635
λo3 (r. sonic) 2.347 2.352
wave position errorspo
1 (l. sonic) 0.972 0.226po
2 (contact) 1.539 0.312po
3 (r. sonic) 0.785 0.202
Table 2.8: Case 1. The contact-shock interaction (step up). Errors for outputwave strengths, wave widths and wave position. Comparison of under resolvedsimulation and prediction.
variable \ error Simulation Predictionmean wave strengths
ωo1 (l. sonic) 0.721 0.712
wave strength errorsVar ωo
1 (l. sonic) 0.0018 0.0018wave width errors
λo1 (l. sonic) 1.718 1.871
wave position errorspo
1 (l. sonic) -0.401 -0.092
Table 2.9: Case 2. The shock crossing equal shock (wave reflection) interaction.Errors for output wave strengths, wave width and wave position. Comparisonof under resolved simulation and prediction.
39
variable \ error Simulation Predictionmean wave strengths
ωo1 (l. sonic) 0.523 0.514
ωo2 (contact) 0.669 0.705
ωo3 (r. sonic) 0.318 0.315
wave strength errorsVar ωo
1 (l. sonic) 0.0009 0.0010Var ωo
2 (contact) 0.0013 0.0013Var ωo
3 (r. sonic) 0.0005 0.0004wave width errors
λo1 (l. sonic) 1.964 2.000
λo2 (contact) 4.606 4.928
λo3 (r. sonic) 2.169 3.029
wave position errorspo
1 (l. sonic) -0.551 -0.103po
2 (contact) 0.301 0.015po
3 (r. sonic) 0.432 -0.131
Table 2.10: Case 3. The contact-shock interaction (step down). Errors foroutput wave strengths, wave width and wave position. Comparison of underresolved simulation and prediction.
40
Chapter 3
Error Analysis for Spherical Geometry
3.1 Introduction
We are concerned with the identification and characterization of solu-
tion errors in spherically symmetric shock interaction problems. This issue
applies to the study of supernova and the design of inertial confinement fusion
(ICF) capsules. In the first case, theory and simulations contain a number of
uncertainties, and comparison to observations is thus not definitive. A sys-
tematic effort to remove some of the uncertainties associated with simulation
will thus be a useful contribution. In the second case of ICF design, concern
over solution accuracy has led to mandates of formal efforts to assure solution
accuracy.
In Chapter 2, we have analyzed shock interactions in a planar geometry
[7, 11], following the general approach to uncertainty and numerical solution
error developed in [22, 23]. Here we focus specifically on complications which
result from spherical geometry. In brief, these are:
1. The solution waves are not of constant strength between wave interac-
41
tions, but evolve approximately according to a power law as a function
of the radius.
2. The solution is not spatially constant between waves.
3. If the solution is not required to be spherically symmetric, the prob-
lem of identifying wave structures as curves or surfaces in 2D or 3D is
introduced.
These are classical problems as far as the solutions are concerned, but
the application of these ideas to the analysis of errors in the solution appears
to be new. The radially dependent strength of spherical waves is discussed
in [40]. The spatial variation of spherical waves is contained in the Guderley
solution [24].
The problem of analysis of errors in numerical solutions is of course central
to numerical analysis. Much of this effort is motivated by other concerns, and
appears not to be directly applicable to the problems we address.
As a technical introduction to this chapter, we study errors of a spherical
shock interaction problem (shown in Fig. 3.1) on uniform radial grid of 100
and 500 cells (with errors determined by reference to a 2000 cell calculation
referred to as the fine grid). The base case for each wave interaction coincides
with the base case assumed for the interactions studied in Sec. 3.2. The trans-
mitted shock, after interaction with the contact, progresses to interact with
(i.e. reflect off) the origin. Subsequently, there are a number of reverberations,
of reflected rarefactions and compression waves, between the contact and the
origin. We use MUSCL [5] as the numerical method; for the comparison of
42
Figure 3.1: Left: space time density contour plot for the multiple wave in-teraction problem studied in this chapter, in spherical geometry. Right: typeand location of waves determined by the wave filter analysis with labels forthe interactions. Here I.R. denotes the inward moving rarefaction.
tracked to untracked solutions of the problem, see [8]. The equation of state is
a γ-law gas with γ = 5/3. The ensemble of 200 initial conditions is defined by
a Latin hypercube variation shock and contact strength by ±10% about a base
case defined (as in Sec. 2.1) by a contact located at 1.5 units from the origin;
an inward moving shock located 2.25 units from the origin, with all constant
states between waves. The initial base case shock strength is M = 32.7 and
the initial base case Atwood number for contact is 0.82.
3.2 The Statistical Numerical Riemann Problem
In this section, we study statistical numerical Riemann problems (SNRP)
in spherical geometry.
43
3.2.1 The Single Propagating Wave
We start with the analysis of the single propagating inward shock in
spherical geometry. The radially dependent strength of convergent spherical
shocks is discussed in [40]. The spatial variation of convergent spherical shocks
is contained in the Guderley solution [24]. The inward moving shocks are not of
constant strength as in a planar geometry, but evolve approximately according
to a power law as a function of the radius. From Whitham’s approximation
approach, we have
M ∝ r−1/n, (3.1)
for cylindrical shocks, and
M ∝ r−2/n, (3.2)
for spherical shocks. Here M is the Mach number of the shock, n = 1 + 2γ
+√2γ
γ−1, and γ is the adiabatic exponent defined as the ratio of two specific heats.
A comparison with the exponents from Guderley’s exact similarity solution is
given in Table 3.1.
Cylindrical Sphericalγ Approximate Exact Approximate Exact
6/5 0.163112 0.161220 0.326223 0.3207527/5 0.197070 0.197294 0.394142 0.3943645/3 0.225425 0.226054 0.450850 0.452692
Table 3.1: Comparison of the exponents from the approximate and the exactsimilarity solutions for an inward propagating spherical shock wave.
We also have a similar approximate power law for the shock velocity
44
Figure 3.2: Left. Mach number vs. radius for a single inward propagatingshock. Right. The same data plotted on a log-log scale.
[12, 34]. Fig. 3.2 shows the exponential divergence of the shock strength (here
characterized by the Mach number) at r → 0. The accuracy is amazing
in view of the simplicity of the approximate theory. The figure shows that
converging shocks are reacting primarily with the geometry, as assumed in the
approximate theory, and are affected very little by further disturbances from
the source of the motion; the strength of the initial shock enters only through
the constants of proportionality in (3.1) and (3.2). This is not true for outward
moving shocks. They slow down due to both the expanding geometry and to
the continuing interaction with the flow behind. From Fig. 3.3, however, we
find that the strength of an outward moving shock also follows a power law
which is similar to (3.2) but with a modified exponent, after the radius of the
outward moving shock is three times the initial radius. To develop a model for
shock wave propagation which has a smaller pre-asymptotic regime, we allow
45
two distinct exponents,
M ∝
ra1
ra2
r0 ≤ r ≤ 3r0
r ≥ 3r0.(3.3)
Here we choose a1 = −0.4, a2 = −1.0 for γ = 1.67, and r0 is the initial shock
radius.
Figure 3.3: Left. Mach number vs. radius for an outward moving shock wavestarting at different radii r0. Right. The same data plotted on a log-log scale;the dashed lines in this plot represent the power law model (3.3).
We also study the single propagating contact (step up and step down
cases). Fig. 3.4 shows the contact width wc ∼ cct1/5 growing from 2 to 5 cells
with a rate asymptotically proportional to t1/5. We found that the step up
contact and the step down contact have the same behavior.
46
Figure 3.4: Ensemble mean contact width for a single propagating contact.We record the width in units of ∆x. The standard deviation is also plotted,as the points to the extreme left.
3.2.2 The Shock Contact Interaction
We study the wave strength, speed, width and position errors after a
wave interaction.
We begin with the analysis of the initial shock contact SNRP at the
ensemble averaged level. We present the linear model coefficients in Table 3.2,
with ±10% variation for the initial contact strength and ±5% variation for
the initial shock strength (consistent with ±10% variation in pressure ratio
as used in Chapter 2, [11]) about the base case. According to the analysis
of Sec. 3.2.1, the strength of this initial inward shock is not constant, and is
increasing as it moves toward the origin. We use the power law M = Cr−2/n to
47
estimate the initial shock strength at the interaction time and use this quantity
represented by the variable C as the input shock strength in the modeling. The
input contact width has been set to zero, as part of the specification of this
SNRP.
variable \ coef const ωi1 ωi
2 model error(contact) (l. sonic) STD STD/ωo
wave strengths (100 cells)ωo
1 (l. sonic) -33.353 19.521 2.501 0.860 0.954%ωo
2 (contact) 0.374 0.200 0.0003 0.042 7.650%ωo
3 (r. sonic) 3.568 0.402 -0.045 0.009 0.463%wave strength errors (100 cells)
δo1 (l. sonic) -2.039 -3.200 -0.01 0.157 0.174%
δo2 (contact) 0.236 0.016 -0.002 0.021 3.825%
δo3 (r. sonic) 0.053 0.003 -0.001 0.0008 0.041%
wave width errors (100 cells)λo
1 (l. sonic) 1.675 0.305 0.017 0.085λo
2 (contact) 7.093 0.482 -0.146 0.239λo
3 (r. sonic) 2.829 0.302 -0.024 0.107wave position errors (100 cells)
po1 (l. sonic) -0.247 0.242 0.005 0.009
po2 (contact) 0.643 0.065 -0.011 0.192
po3 (r. sonic) -0.042 0.062 0.004 0.009
Table 3.2: The SNRP shock contact interaction. Expansion coefficients foroutput wave strengths, wave strength errors, wave width errors and wave po-sition errors (linear model) for the initial shock contact interaction. Here thebase case input contact wave width is zero. The final columns refer to dif-ference between the linear model (2.2) and the exact quantity. The errors inrows 4-12 refer to the difference between the numerical solution on 100 cellsand the exact solution using 2000 cells.
To read Table 3.2, we note that the first (wo1) row (labeled in the table
as wo1 (l. sonic)) lists coefficients α1,J for J = (0, 0), J = (1, 0), etc. These
coefficients are determined by a least squares algorithm that minimizes the
48
expected, or mean error over the ensemble, in comparing the linear predic-
tions to the exact solution of the Riemann problem. The last two columns
describe errors in the model (2.2). The presence of outliers was monitored and
the ensemble L∞ norm determined (results not tabulated); occasional outliers
indicate non Gaussian statistics. The model error is defined as (predicted - ex-
act) where exact is the result of the simulation and predicted is the value given
by the finite polynomial (linear) model (2.2). The column STD is the standard
deviation of (predicted - exact). Note that the STD errors, as defined, are di-
mensionful. To aid in interpreting the error magnitudes, we present in a final
column (labeled STD/wo) the standard deviation of the error in the model
divided by the mean value of the variable predicted. This column represents
a fractional (dimensionless) error in the model.
According to the analysis of Sec. 3.2.1, the strength of the output inward
moving shock is modeled as Cr−2/n. This formula is accurate after some time,
and the Table 3.2 entry is wo1 = C in this formula. We form a linear model for
this constant in this expression in Table 3.2. We find very small errors in the
exponent, not tabulated here. We developed a model (3.3) for the strength of
the output outward moving shock in Sec. 3.2.1. Here in our study, we are only
concerned with the first formula in (3.3). The entry wo3 in the table represents
the coefficient multiplying the power term.
The three variable (λ) rows in Table 3.2 represent wave width errors.
The standard deviation for this quantity is about 10% of the mean value,
indicating that the error model is (on the whole) satisfactory, and that the
shock wave widths are not (mostly) fluctuating greatly. The inward moving
49
Figure 3.5: Left: ensemble mean inward/outward moving shock and contactwidths after a shock contact interaction. Right: ensemble mean shock andcontact position errors as a function of time, expressed in grid units. Theassociated standard deviations are extremely small, not shown in the plots.In the legend, C. denotes the contact while I.S. and O.S. are the inward andoutward moving shocks.
shock width decreased about 10% relative to the wave width at the interaction
time, while the outward moving shock width increased about 10%. See Fig. 3.5,
left frame. The contact width is modeled as cct1/5 where both the width and t
are expressed in mesh units. The Table 3.2 entry λo2 = cc in this formula. We
form a linear model for this constant in this expression in Table 3.2.
We also study the wave position errors. Fig. 3.5, right frame, shows the
position errors as a function of time. The entries in the wave position rows
of Table 3.2 present those errors, given in mesh units. All position errors are
subgrid. The standard deviations are smaller than the means, indicating that
50
the errors are basically deterministic.
All solution errors are sensitive to the grid spacing, taken to be 100
computational cells in Table 3.2-3.4. This sensitivity is not extreme. For
example, if the 100 cell model is used to analyze the 500 cell data, the model
errors (STD) approximately double, but remain small.
3.2.3 Shock Reflection at the Origin
Here we study the reflection of the shock off the origin. According to the
analysis of Sec. 3.2.1, the input inward moving shock has infinite strength at
the origin. We used the strength at the radius r = 1 as the initial state and the
input shock strength in the modeling process. We study the wave strength,
wave strength errors, wave width and wave position errors. See Table 3.3.
variable \ coef const ωi1 model error
(l. sonic) STD STD/ωo
ωo1 (r. sonic) -242.394 5.606 1.137 0.468%
δo1 (r. sonic) -3.27 0.031 0.112 0.045%
λo1 (r. sonic) 1.221 0.018 0.099
po1 (r. sonic) 0.474 0.001 0.012
Table 3.3: The SNRP defined by the shock reflection at the origin. Expansioncoefficients for output wave strengths, wave strength errors, wave width errorsand wave position errors (linear model) for input variation ±10%.
We found that the Mach number of the outward moving shock (reflected
shock) was essentially independent of the input variation in Mach number. To
explain this phenomena, we recall that the ambient state ahead of the outward
moving reflected shock is an incoming continuously variable flow. The sound
51
speed ahead of this flow is affine linearly dependent on the strength of the
incoming shock wave, as is the shock speed of the reflected outward moving
shock wave. Thus the outward moving Mach number, as a ratio of two quan-
tities varying affine linearly with the incoming shock strength, has a fractional
linear form in the incoming wave strength. A simple calculation shows that
the variation in the outward moving shock Mach number Mo contains the fac-
tor (1−Mo) and since Mo ≈ 1.2, this small factor suppressed variation in Mo
as a function of Mi, the Mach number of the incoming shock. Thus the Mach
number is not a good measure for the outward moving shock strength. We
choose the pressure behind the reflected shock instead as ωo1 in Table 3.3. The
pressure also follows the power law. The large entries in this row result from
the fact that the (dimensional) pressure (ωo1) is much larger in pressure units
than the Mach number (ωi1).
3.2.4 The Contact Reshock Interaction
After reflection from the origin, the transmitted lead shock wave re-
crosses the deflected contact. The outgoing waves from this interaction consist
of a rarefaction wave propagating toward the origin, a contact and a shock
propagating outward. The region inside of the outward propagating shock, on
both sides of the contact is not piecewise constant, but contains an inward
propagating compression, which eventually breaks to form an inward moving
shock, reaching the origin at interaction 4. This inward moving compression
is generated from the geometrically caused weakening of the outward moving
shock, and is a well recognized aspect of spherical shock wave dynamics. The
52
variable \ coef const ωi1 ωi
2 model error(r. sonic) (contact) STD STD/ωo
wave strengths (100 cells)ωo
1 (l. sonic) 0.097 -0.108 0.436 0.031 13.305%ωo
2 (contact) 0.103 -0.192 1.168 0.007 1.116%ωo
3 (r. sonic) 0.988 0.195 -0.225 0.003 0.262%wave strength errors (100 cells)
δo1 (l. sonic) -0.291 0.161 -0.468 0.017 7.296%
δo2 (contact) -0.067 0.142 -0.125 0.006 0.957%
δo3 (r. sonic) -0.030 0.107 -0.0004 0.001 0.087%
wave width errors (100 cells)λo
1 (l. sonic) 9.776 -6.372 5.091 0.484λo
2 (contact) 1.903 0.156 -0.677 0.534λo
3 (r. sonic) 4.088 -1.401 1.549 0.168wave position errors (100 cells)
po1 (l. sonic) 4.782 -3.602 2.372 0.379
po2 (contact) -0.453 0.409 -0.054 0.177
po3 (r. sonic) -0.199 -0.685 3.213 0.052
Table 3.4: The SNRP contact reshock interaction. Expansion coefficientsfor output wave strengths, wave strength errors, wave width errors and waveposition errors (linear model).
shock and the rarefaction interact, and eventually the rarefaction disappears
in this interaction. Here we only follow the waves through the output of in-
teraction 3, and thus avoid much of this interaction. Specifically, we focus on
the inward moving rarefaction and not the inward moving shock. We study
the wave strength, wave strength errors, wave width and wave position errors
resulting from interaction 3. See Table 3.4.
According to the analysis of Sec. 3.2.1, this is a step down interaction
and the contact width is modeled as cct1/5 where both the width and t are
expressed in mesh units. We form a linear model for the coefficient cc in this
53
expression in Table 3.4. The rarefaction width has the form constant + rate
× time. The entry λo1 refers to the constant, which gives an offset for the
centering of the rarefaction wave. This entry is expressed in mesh units.
3.3 Composite Shock Interaction Problems
The main point of this section is to formulate and validate the multipath
scattering formula
S =∑
B⊂V(G)
SB =∑
B⊂V(G)
∫ ∏v∈B
IvdωB . (3.4)
that we developed in Sec. 2.3 ([11]) for analysis of errors. We analyze errors
at the output to interaction 3 directly, comparing the 100 mesh and 500 mesh
simulation to a 2000 mesh, fine grid simulation, here taken as a substitute for
the exact solution. These errors are compared to those generated by adding up
and propagating errors from the input data and from each of the interactions
1 to 3, using the multipath scattering formula. Thus, for example, a position
error as input to interaction 1 is translated geometrically to a position error
for the output to interaction 1 via simple geometric considerations as in [11].
This error is propagated to an input error for interaction 2 through solutions
of radial differential equations. Propagation continues, and yields an error at
the output to interaction 3. See Table 3.5. The wave strength rows present the
result of initial uncertainty propagated to the output of interaction 3 as well
as the accumulation of solution errors. The multipath scattering formula gives
reasonable prediction of error magnitudes in all cases except the wave position
54
errors for the under resolved (100 mesh) simulation. We see that the created
numerical solution errors are important. We also find that a major portion
of the created numerical solution errors come from the second interaction, the
shock reflection interaction. A detailed study of these errors and their relative
importance will be presented in the next section.
variable \ error Simulation Prediction Simulation Prediction100 vs. 2000 mesh 500 vs. 2000 mesh
wave strength errors and propagated initial uncertaintiesδo1 (l. sonic) 0.04±2(0.03) 0.03±2(0.02) 0.01±2(0.02) 0.009±2(0.01)
δo2 (contact) 0.14±2(0.05) 0.12±2(0.02) 0.03±2(0.01) 0.03±2(0.008)
δo3 (r. sonic) -0.02±2(0.02) -0.02±2(0.01) -0.006±2(0.005) -0.007±2(0.004)
mean wave width errors mean wave width errorsλo
1 (l. sonic) 3.04 2.83 2.63 2.72λo
2 (contact) 5.36 6.11 5.56 6.08λo
3 (r. sonic) 2.71 3.04 2.92 2.98mean wave position errors mean wave position errors
po1 (l. sonic) 1.25 0.23 0.12 0.18
po2 (contact) 0.43 0.06 0.05 0.04
po3 (r. sonic) -0.73 -0.15 -0.08 -0.11
Table 3.5: Predicted and simulated errors for output wave strengths, wavewidths and wave positions, output to interaction 3. The inward rarefactionand contact strengths are expressed dimensionlessly as Atwood numbers. Theoutward shock strengths are in the units of Mach number. The width andposition errors are in mesh units. The wave strength errors are expressed asmean ± 2σ where σ is the ensemble STD of the error/uncertainty.
3.4 Error Decomposition
In this section, we will study the relative magnitude of the input uncer-
tainty vs. the errors created within the numerical solution. In more detail,
we wish to understand the contribution of each wave interaction to the errors
55
observed at the end of the simulation. Here, the multipath integral formula is
used assuming independence of errors from different sources.
In Fig. 3.6 we illustrate the distinct terms contributing to multipath scat-
tering formula for analysis of the errors. Each number labelled line segment
is a single term (error propagating path), for the error associated with the
output to interaction 3, in which the shock wave reflected from the origin re-
crosses (re-shocks) the contact. The first two paths indicate the uncertainty
originating with the initial conditions, i.e. with the choice of the ensemble.
This uncertainty propagates through two distinct paths, illustrated by the first
two black lines of Fig. 3.6, to reach the interaction site 3. The first one follows
the shocks, the transmitted shock from the interaction 1 to the origin reflected
shock and back to the contact. The second one follows the the contact from
the lead interaction 1, along the contact until it is reshocked at interaction
3. Next we find two gray lines that represent the errors originating during
the interaction 1, and propagating to the output of 3 through the same two
routes. Finally, we find two paths giving the errors that arise at the shock
origin reflection (interaction 2) and propagate to 3 and in the final path, those
arising during interaction 3 directly.
In Figs. 3.7, 3.8, we present three pie charts representing fractional con-
tribution from each of the six interactions to the error variance for the inward
rarefaction, contact and outward shock, respectively, as output to interaction
3. Note that the two associated with input uncertainty are hatched and the
others are solid gray scales. From these charts, we can infer the relative im-
portance between the input uncertainty and the solution error and determine
56
the contribution of each interaction to the total error variance. We see that for
a 500 cell grid, the dominant error comes from the initial uncertainty, while
for the 100 grid over 75% of the error arises within the numerical simulation.
We also show the contributions of each interaction to the mean value of
the final total error. See Table. 3.6. We only show the values correspond-
ing to Diagrams 3 to 6, as the contribution of the first two diagrams (input
uncertainties) is observed to be zero.
Wave Diagram I.R. C. O.S.Number 100 500 100 500 100 500
3 0.10 -0.01 -0.01 0.001 0.09 -0.0094 0.05 0.009 0.1 0.02 -0.02 -0.0045 -0.05 -0.005 0.006 0.0006 -0.04 -0.0046 -0.07 0.015 0.03 0.01 -0.05 0.01
Total Prediction 0.03 0.009 0.12 0.03 -0.02 -0.007Total Simulation 0.04 0.01 0.14 0.03 -0.02 -0.006
Table 3.6: The contribution of each interaction to the mean value of the totalerror in each of three output waves at the output to interaction 3, for 100 and500 mesh units. Units are dimensionless and represent the error expressed asa fraction of the total wave strength. The last two rows compare the totalof the mean error as given by the model to the directly observed mean error.The columns I.R., C., and O. S. are labeled as in Fig. 3.1, Right frame.
57
Figure 3.6: Schematic graph showing all six wave interaction contributionsto the errors or uncertainty in the output from a single Riemann solution,namely the reshock interaction (numbered 3 in the right frame of Fig. 3.1)of the reflected shock from the origin as it crosses the contact. The numberslabeling the circles refer to the Riemann interactions contributing to the error.The numbers labeling the line segments refer to the different error propagatingpaths.
58
(a) Inward Rarefaction
(b) Contact
(c) Outward Shock
Figure 3.7: Pie charts showing the contribution of each wave interaction dia-gram to the error variance of the wave strength at the output of interaction 3,for a solution using 500 mesh units.
59
(a) Inward Rarefaction
(b) Contact
(c) Outward Shock
Figure 3.8: Pie charts showing the contribution of each wave interaction dia-gram to the error variance of the wave strength at the output of interaction 3,for a solution using 100 mesh units.
60
Chapter 4
Conclusions
We have several main conclusions from the error study.
• We see that a very simple model of solution error is sufficient for the
study of (at least the present instance of) a highly nonlinear problem.
The error is linear in the input wave strengths.
• A composition law for combining errors and predicting errors for compos-
ite interactions on the basis of an error model of the simple constituent
interactions has been formulated and validated. For spherically symmet-
ric shock physics problems, the main new difficulties encountered were
the non-constancy of the solution between interaction events and the
non-constancy of waves and errors between interactions. For a planar
geometry, the errors are constant between interactions, while for a spher-
ical geometry, the errors grow (if the wave which carries them is moving
inward) by a power law in the radius. Similarly outward moving waves
and their errors weaken by a power law.
61
• For planar case, we find that although our formalism allows for statistical
errors in the ensemble that in fact, the dominant part of all errors (ex-
cluding position errors) studied were deterministic, in the sense that the
ensemble mean error dominated the ensemble standard deviation. For
spherical case, the composition model applied to construct the variance
of the error in the wave strength, generally understates the STD by a
factor generally between 1.5 and 2, for causes not presently identified.
Using the model, the total error is a sum of six terms, each corresponding
to a pattern of wave interactions and transmissions. Of these diagrams,
two correspond to initial error, following different transmission patterns,
and four correspond to errors created within the solution and trans-
mitted to the output of the reshock interaction, where the errors are
analyzed. We see that for a 500 cell grid, the dominant error comes from
the initial uncertainty, while for the 100 grid over 75% of the error arises
within the numerical simulation. We could conclude that for coarse grid
simulations, there exists increased importance of created errors.
• For planar case, the wave strength uncertainty which is dominated by
input uncertainty (i.e. the definition of the ensemble), is virtually un-
changed between the highly resolved and the under resolved simulations.
While for spherical case, this is not true. We concluded that for under
resolved spherical simulations, there exists increased importance of cre-
ated errors. The wave width errors are both expressed in grid units, and
are comparable between the two levels of resolution. The wave width
errors evidentially have a rapid relaxation to their asymptotic value.
62
• The primary solution errors created by the simulations are the wave
position errors in the under resolved simulations, on the order of a mesh
spacing. These errors are a transient phenomena, but become frozen
into the calculation as new interactions occur before the transient errors
have diminished. The wave position errors have a slow relaxation to
asymptotic values.
• We find that the wave filter performs well as a diagnostic tool, but that its
limitation (in its present version) lies in assuming well separated waves.
Thus we are limited in the degree of under resolution that we can analyze,
in that all waves must be at least partially separated from one another
before entering into a new interaction.
To the extent that a more detailed modelling of these errors is important,
a more accurate model that includes transient effects will be important. Even
with these limitations, the methods and results appear to be promising, and
should be extended to less idealized problems.
These conclusions are established only under several simplifying assump-
tions, namely restriction to one spatial dimension, use of a simplified (gamma
law gas) equation of state, and consideration of only one numerical method.
Further studies are needed to determine the extent that these conclusions have
a general validity.
63
Chapter 5
Future Work
5.1 2D Shock Wave Interactions in Perturbed Spherical
Geometries
For the future work, we address the much more difficult question of the
perturbed interface problem, with the ensuing instability growth and chaotic
flow. Before we can apply the above methods of statistical error analysis, we
require a numerical solution procedure which is O(1) correct. This simply
stated and seemingly elementary requirement has proven to be surprisingly
difficult for the scientific community. Our present results apply to a planar
geometry. After the first correct RM simulation (to achieve agreement with
experimental data) by FronT ier [27, 29], a three way code comparison, with
experimental data from laser acceleration and a theoretical model all achieved
agreement for a single mode 2D RM instability [28]. This result is very encour-
aging, but the mesh resolution used to achieve it was not, in view of the desire
to compute the much more difficult instabilities in 3D and for fully chaotic (as
64
opposed to single mode) flows. For the related RT chaotic instability in 3D,
extensive studies have shown that most simulation codes compute an insta-
bility growth rate which is below the experimental value by about a factor of
2, while the FronT ier values [9, 10, 13] are consistent with the experimental
range of values, but on the high side. For FronT ier, we could introduce sur-
face tension or mass diffusion to lower the instability growth rate. While for
other simulation codes, adding long wave noise in the initial condition would
be one way to raise the instability growth rate.
5.1.1 Single Mode Perturbed Interface
In Fig. 5.1, we show the density plot for a spherical implosion with a single
mode perturbed interface. We will start with the solution convergence of direct
numerical simulations (DNS) of single mode flow through mesh refinement,
regardless of different numerical algorithms. The convergence should be on
the point-wise sense. Meanwhile, we will determine the solution sensitivity to
mesh size, algorithm, mass diffusion and etc.
We define the L1-error of two different mesh simulations. Suppose ρc(x, t)
is the discrete density field from a coarse grid simulation (as for example
computed on a fixed size Eulerian mesh with grid size measured by h) and
ρf (x, t) is a fine gird solution density. we define the time dependent L1-error
on the computational domain Ω as:
‖ρc − ρf‖L1(t) =
∫Ω
|ρc(x, t) − ρf (x, t)|dx. (5.1)
65
Figure 5.1: Density plot for a spherical implosion simulation with a perturbedinterface (single mode). The grid size is 200 × 200.
For the simulations described in this chapter, the computational domain is
given by Ω = x :√
x · x ≤ Rmax. Since we are assuming axisymmetry for
our flow, the above integral can computed by the formula:
‖ρh − ρf‖L1(t) =
∫Ω
|ρh(r, z, t) − ρf (r, z, t)|rdrdz, (5.2)
5.1.2 Chaotic Mixing
Chaotic flows display a wealth of detail which is not reproducible, nei-
ther experimentally nor in simulations. Generally speaking, this detail is not
relevant, and fortunately, only the statistical averages of the detail are of im-
portance. Thus direct numerical simulation (DNS) of mix, as discussed in
the previous section, gives more information than is needed, and information
66
which in detail cannot be reproducible. Since we really want the averages of
the DNS simulations, the natural question is to find averaged equations which
will compute the averaged quantities directly, without use of the difficult in-
termediate DNS step.
Figure 5.2: Density plot for a spherical implosion simulation with a chaoticperturbed interface (multiple modes). The grid size is 200 × 200.
In Fig. 5.2, we show the density plot for a spherical implosion with a
chaotic perturbed interface. We will analyze the averaged quantities. The
averaged DNS solution should converge under mesh refinement and under
ensemble size approaching infinity. Then we will compare the averaged DNS
solution to the solution of averaged equations [31] (see also earlier work [20, 30]
and references cited there).
Averaged equations arise in many areas of science. Generally, when the
original equations are nonlinear, or when the coefficients of a linear term are
67
to be averaged, lengthy discussions of how to formulate the averaged equations
ensue. The issue is that nonlinearities do not commute with averaging, so the
average of a nonlinear function is not equal to the function evaluated at the
average value of its argument. In addition, the phenomena at a physics level
are much richer, as the averages depend on the averaging length scale. We
wish to average over each phase, and end up with multi-phase flow equations.
The nonlinear closure terms will then reflect the forces, etc., exerted between
the two phases.
From the comparison of averaged DNS solution and solution of the aver-
aged equation, we will be able to examine the adequacy of closures and com-
pose distinct forms of averaging. Also, we will analyze the numerical errors in
solution of averaged equations.
68
Chapter 6
Appendix
6.1 Complete List of Ten Riemann Problems (Planar
Geometry)
For each of the 10 Riemann problems of Fig. 2.2, Right, we vary the wave
strength and for contacts only, we vary the wave width. However, when solved
using the (idealized) Riemann solver, the wave widths are all set to zero. Three
variables out of the nine variables which define three reference states (for two
waves) are not varied in this study. We have two criteria in selecting the
reference variables to hold fixed. If one of the states has a reference ambient
velocity, for example a velocity v = 0 for a state near a wall, we want to
preserve this property and freeze this velocity. For the pressure and density
values, we generally freeze those on the smaller side of the waves, as this gives
a more meaningful variation of the state, uniformly specified as ±10% of the
wave strength, as defined in Sec. 2.2.
Here, we present error models for cases 4-10 in this multiple wave inter-
69
action problem, to complete the analysis of cases 1-3 in Chapter.2.
Case 1: Lead shock interacts with contact
The mid state is held fixed, and the two wave strengths are varied.
Figure 6.1: Problem 1: Shock-contact (step up)
Case 2: Transmitted shock reflects off of wall
The right state is held fixed and the left state is varied.
Figure 6.2: Problem 2: Shock-wall interaction
70
Case 3: Shock reflected off wall recrosses the contact
The right state velocity v = 0.0 is fixed and the left state densities and
pressures are held fixed.
Figure 6.3: Problem 3: Contact-shock (step down)
Case 4: Reflected rarefaction from case 3 reflects off of
wall
Here we study the reflection of the rarefaction off the wall, a special case
of the rarefaction crossing rarefaction interaction. The right state velocity
v = 0.0 is fixed and the left state densities and pressures are held fixed. When
modelled as a SRP, the input rarefaction wave width is set to zero. When
modelled as a SNRP, the input rarefaction wave width is an input parameter.
Similar comments apply to the most of the later cases.
We have one input and one output wave width parameter, both for the
rarefaction. We assume that at interaction location the input rarefaction and
the output rarefaction have the same width. The rarefaction width has the
form
constant + initial width + rate × time (6.1)
71
Figure 6.4: Problem 4: Rarefaction-wall
Here, initial width means the input rarefaction width, which would occur at
the interaction time without the influence of the interaction. We find very
small errors in the rate, not tabulated here. The entry λo1 in Table 6.1 refers
to the constant in formula (6.1), which gives an offset for the width of the
rarefaction wave. This entry is expressed in mesh units. We form a linear
model for this constant in Table 6.1.
variable \ coef const ωi1 error
(r. sonic) L∞ STDωo
1 (l. sonic) 0.146 0.657 0.181% 0.0003λo
1 (l. sonic) 3.832 2.091 5.056% 0.096po
1 (l. sonic) 0.117 -0.008 15.56% 0.006
Table 6.1: Case 4. The SNRP defined by the crossing of two rarefactions.Expansion coefficients for output wave strengths (linear model) for input vari-ation ±10%.
Case 5: Rarefaction reflected off of wall crosses contact
After reflection from the wall, the transmitted rarefaction wave re-crosses
the deflected contact. The right state velocity v = 0.0 is fixed and the left
72
state densities and pressures are held fixed.
Figure 6.5: Problem 5: Contact-rarefaction
We have two input wave width parameters, one each for the contact and
the rarefaction, and three output wave width parameters, one each for the left
rarefaction, the contact and the right compression wave. According to the
analysis of single isolated waves, the output contact width is bounded after
some 100 time steps. The Table 6.2 entry λo2 refers to this bounded width. We
assume that at interaction the input rarefaction and the output rarefaction
have the same width. The rarefaction width has the form (6.1). Here, initial
width means the width, which eon would have at the interaction time without
the influence of the interaction. We find very small errors in the rate, not
tabulated here. The entry λo1 refers to the constant in formula (6.1), which
gives an offset for the width of the rarefaction wave. This entry is expressed in
mesh units. We form a linear model for this constant in Table 6.2. Similarly,
the compression wave width has the form (6.1). Here the rate has a negative
sign. We find very small errors in the rate, not tabulated here. The entry λo3
refers to the constant in formula (6.1), which gives an offset for the width of
the compression wave. This entry is expressed in mesh units. We form a linear
73
model for this constant in Table 6.2.
variable \ coef const ωi1 ωi
2 error(contact) (l. sonic) L∞ STD
ωo1 (l. sonic) 0.020 -0.073 0.697 0.33% 0.0002
ωo2 (contact) 0.198 1.076 -0.716 0.30% 0.0006
ωo3 (r. sonic) -0.029 0.107 0.295 0.76% 0.0003
λo1 (l. sonic) 4.613 0.018 7.378 10.5% 0.206
λo2 (contact) -3.480 -0.514 19.137 11.8% 0.091
λo3 (r. sonic) 6.260 0.777 3.812 3.09% 0.111
po1 (l. sonic) 0.019 0.205 0.524 21.21% 0.023
po2 (contact) -0.12 0.023 0.779 0.783% 0.023
po3 (r. sonic) 0.186 0.056 0.410 25.23% 0.027
Table 6.2: Case 5. The SNRP defined by the contact rarefaction interaction.Expansion coefficients for output wave strengths (linear model) for input vari-ation ±10%.
Case 6: Reflected shocks from interactions 1 and 3 over-
take
In this case, the overtake of reflected shocks from interactions 1 and 3 is
studied. The left state is held fixed and the two wave strengths are varied.
Figure 6.6: Problem 6: Shock-shock overtake (two waves of the same family)
74
The two input shocks are both left moving forward shocks, which are
produced by interactions 1 and 3, respectively. One left moving forward shock,
one contact and one right moving backward shock are produced. We ignore
the output backward shock because it is too weak to be recognized by the
filter program. Therefore, the position error po3 and the width error λo
3 are not
presented here.
However, with an improved post-processing program, we can still get all
input and output gas states, so all input and output strengths are studied
here.
From the analysis of the section 2.2.1, the contact width will be modeled
as wc ∼ cct1/3, where the coefficient cc is affected to Mach number, and the
t denotes the time steps counted from the interaction time. Thus we model
this case in the same manner as the step down problem. This also is showed
by the Fig. 6.7, in which we plot the time step axis on a log scale to show
this relationship. The plot starts from 51, because the contact widths near
interaction time are unstable, and do not follow this scaling law. We observe
an increase in width from 4 cells to 9 over 1000 time steps.
In this plot there is a noticeable small disturbance after 1000 time steps,
beyond an otherwise good linear relation. That disturbance is due to an addi-
tional wave interaction with the contact at that time, which therefore affects
the contact width after 1000 time steps. This interaction is not presented here
since it is not one of the major interactions, but still it can be seen in Figure
10 in the paper [11].
Thus, in Table 6.3 we interpret λo2t
1/3 or λo2 as the output contact wave
75
Figure 6.7: Because the width is entirely grid related, we record width in unitsof ∆x and time in units of the number of time steps.
width, also in grid units, depending on the Mach number. For λo1, it is the
shock wave width and independent of time steps. And pak and ωa
k (k = 1, 2, 3
and a = i, o) represent position errors and wave strengths as usual.
The width of the left most input forward shock, which is from interaction
1, is 1 grid unit. The width of the other input forward shock, which is from
interaction 3, is 2 grid units.
The output position errors are computed assuming zero input position
error to see their relationship with the input wave strengths, according to the
formulae introduced by Section 2.3.2.
76
variable \ coef const ωi1 ωi
2 error(l. sonic) (l. sonic) L∞ STD
ωo1 (l. sonic) 0.033 0.375 1.110 0.02% 0.0001
ωo2 (contact) -0.028 0.075 0.149 0.91% 0.0002
ωo3 (r. sonic) -0.018 0.014 0.056 0.57% 0.0000
λo1 (l. sonic) 2.050 -0.248 -0.224 9.57% 0.078
λo2 (contact) 0.092 0.278 1.418 1.71% 0.007
po1 (l. sonic) 0.132 -0.030 0.007 12.92% 0.009
po2 (contact) -0.026 0.010 0.233 32.06% 0.009
Table 6.3: Case 6. The SNRP defined by the shock shock overtake (two wavesof the same family). Expansion coefficients for output wave strengths (linearmodel) for input variation ±10%.
Case 7: Compression wave reflected from interaction 5
reflects off of wall
Here we study the reflection of the compression off the wall, a special case
of the compression crossing compression interaction. The right state is held
fixed.
Figure 6.8: Problem 7: Compression-wall
Similar to the previous cases, the compression wave width has the form
(6.1). Here, the initial width denotes the compression width, which would
77
occur at the interaction time without the influence of the interaction. The rate
has a negative sign. and, we find very small errors in the rate, not tabulated
here. The entry λo1 refers to the constant in formula (6.1), which gives an
offset for the width of the compression wave. This entry is expressed in mesh
units. We form a linear model for this constant in Table 6.4. We find less
than 0.001% density oscillation near wall. So, there is no wall error in the
compression wall interaction.
variable \ coef const ωi1 error
(r. sonic) L∞ STDωo
1 (l. sonic) -0.026 1.127 1.450% 0.00061λo
1 (l. sonic) -0.893 -2.323 31.41% 0.107po
1 (l. sonic) 0.034 -0.223 2028% 0.026
Table 6.4: Case 7. The SNRP defined by the crossing of two compressions.Expansion coefficients for output wave strengths (linear model) for input vari-ation ±10%.
Case 8: Compression wave from wall reflection crosses
contact
After reflection from the wall, the transmitted compression wave re-
crosses the reflected contact. The right state velocity v = 0.0 is fixed and
the left state densities and pressures are held fixed.
We have two input wave width parameters, one each for the contact and
the compression, and three output wave width parameters, each for the left
compression, the contact and the right rarefaction wave. The entry λo1 in
78
Figure 6.9: Problem 8: Contact-compression
Table 6.5 refers to the stable width of the transmitted compression wave after
it is changed to a shock wave.
variable \ coef const ωi1 ωi
2 error(contact) (l. sonic) L∞ STD
ωo1 (l. sonic) 0.030 -0.039 0.601 3.46% 0.0008
ωo2 (contact) -0.043 0.973 0.442 0.71% 0.0011
ωo3 (r. sonic) -0.026 0.034 0.402 4.29% 0.0007
λo1 (l. sonic) 3.298 0.116 -1.510 13.2% 0.080
λo2 (contact) -0.240 -0.081 -4.584 3480% 0.432
λo3 (r. sonic) 3.758 -1.265 9.379 19.7% 0.273
po1 (l. sonic) -0.311 1.842 -2.484 137.5% 0.263
po2 (contact) 0.228 -0.230 -0.081 239.4% 0.147
po3 (r. sonic) -0.440 -0.935 2.665 49.3% 0.166
Table 6.5: Case 8. The SNRP defined by the contact compression interaction.Expansion coefficients for output wave strengths (linear model) for input va-ration ±10%.
79
Case 9: Rarefaction reflected from interaction 8 reflects
off of wall
Case 9 is the reflection of the rarefaction off the wall again. We used
the same functional form for the error model as in case 4, but the numerical
coefficients are quite different. The right state velocity v = 0.0 is fixed and
the left state densities and pressures are held fixed.
Figure 6.10: Problem 9: Rarefaction-wall
variable \ coef const ωi1 error
(r. sonic) L∞ STDωo
1 (l. sonic) 0.004 0.938 0.55% 0.0001λo
1 (l. sonic) -0.060 5.290 206% 0.067po
1 (l. sonic) -0.027 -0.965 1133% 0.036
Table 6.6: Case 9. The SNRP defined by the crossing of two rarefactions.Expansion coefficients for output wave strengths (linear model) for input vari-ation ±10%.
80
Case 10: Rarefaction reflected off of wall passes through
contact
The output wave strengths in case 10 are very weak (ωo1 = 0.033, ωo
2 =
0.674, ωo3 = 0.021). The wave filter lacks sufficient precision to allow analysis
of the errors in this case, which because their small size are not important in
any case. The right state velocity v = 0.0 is fixed and the left state densities
and pressures are held fixed.
Figure 6.11: Problem 10: Contact-rarefaction
6.2 Errors in Resolved Calculations
We examine errors in wave strength, wave position and wave width, based
on the graphical expansion given in the Chapter 2. The wave strength errors
are dominated by the transmission of error (or uncertainty) from the initial
conditions. The wave width errors and the wave position errors are dominated
by the created error in the current interaction and the transmission of errors
from previous interactions. In Tables 6.7, 6.8, 6.9, 6.10, 6.11 and 6.12, we
compare the predicted error with the error computed directly, taken from a
81
full solution of the multiple wave interaction problem. The model for the
prediction of the error is satisfactory for those cases: the wave strength and
its errors, the wave width errors.
variable \ error Simulation Predictionmean wave strengths
ωo1 (l. sonic) 0.346 0.348
wave strength errorsVar ωo
1 (l. sonic) 0.0004 0.0003wave width errors
λo1 (l. sonic) 4.568 4.474
wave position errorspo
1 (l. sonic) 0.356 0.244
Table 6.7: Case 4. The crossing of two rarefactions. Predicted and simulatederrors for output wave strengths, wave widths and wave positions.
82
variable \ error Simulation Predictionmean wave strengths
ωo1 (l. sonic) 0.212 0.212
ωo2 (contact) 0.675 0.672
ωo3 (r. sonic) 0.147 0.145
wave strength errorsVar ωo
1 (l. sonic) 0.0002 0.0002Var ωo
2 (contact) 0.0011 0.0014Var ωo
3 (r. sonic) 0.0001 0.0001wave width errors
λo1 (l. sonic) 7.212 7.176
λo2 (contact) 2.417 2.790
λo3 (r. sonic) 7.895 8.101
wave position errorspo
1 (l. sonic) 1.334 0.941po
2 (contact) 0.021 0.010po
3 (r. sonic) -0.159 -0.042
Table 6.8: Case 5. The contact rarefaction interaction. Predicted and simu-lated errors for output wave strengths, wave widths and wave positions.
83
variable \ error Simulation Predictionmean wave strengths
ωo1 (l. sonic) 0.783 0.781
ωo2 (contact) 0.079 0.083
ωo3 (r. sonic) 0.015 0.017
wave strength errorsVar ωo
1 (l. sonic) 0.0016 0.0019Var ωo
2 (contact) 0.000001 0.000005Var ωo
3 (r. sonic) 0.00003 0.00004wave width errors
λo1 (l. sonic) 1.808 1.821
λo2 (contact) 3.947 4.248
wave position errorspo
1 (l. sonic) 0.678 0.427po
2 (contact) 0.797 0.589
Table 6.9: Case 6. The shock shock overtake. Predicted and simulated errorsfor output wave strengths, wave widths and wave positions.
variable \ error Simulation Predictionmean wave strengths
ωo1 (l. sonic) 0.138 0.139
wave strength errorsVar ωo
1 (l. sonic) 0.0001 0.0001wave width errors
λo1 (l. sonic) 8.183 6.662
wave position errorspo
1 (l. sonic) 0.152 0.038
Table 6.10: Case 7. The crossing of two compressions. Predicted and simulatederrors for output wave strengths, wave widths and wave positions.
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variable \ error Simulation Predictionmean wave strengths
ωo1 (l. sonic) 0.086 0.086
ωo2 (contact) 0.675 0.674
ωo3 (r. sonic) 0.053 0.053
wave strength errorsVar ωo
1 (l. sonic) 0.00003 0.00003Var ωo
2 (contact) 0.0013 0.0013Var ωo
3 (r. sonic) 0.00002 0.00002wave width errors
λo1 (l. sonic) 2.843 3.168
λo2 (contact) -1.356 -0.927
λo3 (r. sonic) 12.794 11.381
wave position errorspo
1 (l. sonic) 1.321 0.613po
2 (contact) 0.186 0.078po
3 (r. sonic) -1.526 -0.715
Table 6.11: Case 8. The contact compression interaction. Predicted andsimulated errors for output wave strengths, wave widths and wave positions.
variable \ error Simulation Predictionmean wave strengths
ωo1 (l. sonic) 0.054 0.053
wave strength errorsVar ωo
1 (l. sonic) 0.00002 0.00002wave width errors
λo1 (l. sonic) 11.293 10.013
wave position errorspo
1 (l. sonic) 1.954 0.669
Table 6.12: Case 9. The crossing of two rarefactions. Predicted and simulatederrors for output wave strengths, wave widths and wave positions.
85
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