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Lacuna-based Artificial Boundary Condition And Uncertainty Quantification of the Two-FluidPlasma Model
Eder Sousa1, Uri Shumlak1 and Guang Lin2
1 Computational Plasma Dynamics Lab, University of Washington, Seattle, WA2 Computational Sciences and Mathematics, Pacific Northwest National Laboratory, Richland, WA
Abstract
Modeling open boundaries is useful for truncating extended or infinite simulation domains to regions of greatest interest. However, artificial wave reflections at the boundaries can result for oblique wave intersections.
The lacuna-based artificial boundary condition (ABC) method is applied to numerical simulations of the two-fluid plasma model on unbounded domains to avoid unphysical reflections. The method is temporally
nonlocal and can handle arbitrary boundary shapes with no fitting needed nor accuracy loss. The algorithm is based on the presence of lacunae (aft fronts of the waves) in wave-type solutions in odd- dimensional
space. The method is applied to Maxwell’s equations of the two-fluid model. Placing error bounds on numerical simulations results is important for accurate comparisons, therefore, the multi-level Monte Carlo method
is used to quantify the uncertainty of the two-fluid plasma model as applied to the GEM magnetic reconnection problem to study the sensitivity of the problem to uncertainty on the mass ratio, speed of light to Alfven
speed ratio and the magnitude of the magnetic field initial perturbation.
Lacuna-based Artificial Boundary Conditions 3
INumerical simulation of wave phenomena on unbounded domains often produce unphysical reflections from the boundariesIConsequently, The original infinite domain has to be truncated and special artificial boundary conditions (ABCs) have to be developed
Lacunae are still regions present in wave-type solutions in odd-dimension
spaces.
Introduction
IThe key idea of using lacunae for computations isvery simple:I If the sources of waves are compactly supported in
space and time;I If the domain of interest has finite size;IThen it will completely fall inside the lacuna once a
certain time interval has elapsed since the inceptionof the sources.
IThe lacunae-based ABC is nonlocal in space andtime without loss of accuracy
IThe lacunae-based ABCs are not restrictedgeometrically to any particular shape of the externalartificial boundary
The computational domain and the auxiliary domain overlap by a couple gridcells where the transition multiplier, µ, varies from zero on the interior side to oneon the exterior one. The three steps of ABC implementation:ICalculate the exterior source, Ω(v) , from the interior solution;IReintegration of the exterior solution excluding earlier exterior sourcesICommunicate the exterior evolution with the interior problem ghost cells
3S.V.Tsynkov, ”On the Application of Lacunae-based Methods to Maxwell’s Equations”, JCP 199
(2004) 126-149
Auxiliary source generation
IThe computational domain is advanced using:∂q∂t
+∇ · F(q) = S(q)
IAuxiliary problem: ∂v∂t
+∇ · F(v) = S(v) + Ω(q)
IΩ(q) is the auxiliary source and v = µqIFor non-moving boundaries there is no time
dependent of µ(x), therefore:Ω(q) = µS(q)− S(µq) +∇ · F(µq)− µ∇ · F(q)
The boundary formulation is applied to the Maxwell equations using theWashington Approximate Riemann Plasma Solver (WARPX). The field aremodeled using the Perfectly Hyperbolic Maxwell4 formulation to account for thedivergence corrections,
∂~B∂t
+∇× ~E + γ∇Ψ = 0
1c2∂~E∂t−∇× ~B + χ∇Φ = −µo
∑s
qs
msρs~us
1χ
∂Φ
∂t+∇ · ~E =
∑s
qs
msρs
1γc2
∂Ψ
∂t+∇ · ~B = 0
Quantity being plotted is Bz.
– Interior domain boundary, – Auxiliary domain boundary
Problem setup: a quarter of spherical pulse propagating outwards, where the left
and the bottom boundary conditions are lacunae-based ABC’s. The wave front is
communicated to the auxiliary problems by the auxiliary sources in the transition
region (overlap region between the interior and the auxiliary regions).
Exterior domain re-integration
IThe auxiliary sources drive the problem in theauxiliary domain and guarantees both solutionsmatch in the exterior domain
IThe auxiliary problem is re-integrated every specifiedtime interval and early sources are removed fromcomputation
The following plot are slices of the previous figures at x=1.
Initially the interior problem pulse propagates
through the interior domain.
As the re-integration is preformed the earlier
sources are removed from the auxiliary
domain as they no longer affect the interior
solution.
As the interior pulse enters the transition
region, the auxiliary source is applied to the
auxiliary domain.
The pulse is reintegrated out of the exterior
domain an no reflection are present in the
interior problem.
Conclusion
The lacuna-based ABC’s can correctly simulated unbounded
domains accurately while removing artificial reflections that
otherwise would be present.
4Munz et. al., ”Divergence Correction Techniques for Maxwell Solvers Based on Hyperbolic
Model”, JCP 161, 484-511 (2000)
Uncertainty Quantification
Motivation
IDetermining the region of acceptable results forexperimental and computational is not only desiredbut required
IUncertainty quantification will allow for errorbars to beput into computational results
IThere are numerous sources of uncertainty in theTwo-Fluid plasma model
ITreating all the inputs as stochastic is computationallyexpensive
Introduction
IThe mean square error (MSE)e(PM)2 = V [PMC
ml,N] + (E [Pml − P])2
ITo achieve root MSE less than εI the variance V [PMC
ml,N] = N−1V [Pml] has to be less
than ε2/2 meaning N ≥ ε−2 for the first term (largenumber of samples)
IHigh discretization level: ml ≥ ε−1/α where α is thediscretization convergence rate for the second term
Multi-level Monte Carlo (MMC)5,6
IThe method is based on the multi-grid method as thesolution is obtained from different solutions at differentgrid refinement levels
IThe estimator comes from the same random sample,N, but at different refinement levels, L
PMLml
=L∑
l=0
1Nl
Nl∑i=1
(P iml− P i
ml−1)
IThe multilevel variance V [Yl] = V [Pml − Pml−1]→ 0 asl →∞ which implies that Nl → 1 as l →∞
I It is less costly to achieve an overall RMSE of ε for themultilevel than the standard Monte Carlo
Probabilistic Collocation Method (PCM)
IThe PCM approach based on selecting the samplingpoints and corresponding weights.
ICollocation points in probability space of randomparameters as independent random inputs based ona quadrature formula
IThe solution statistics is estimated using thecorresponding quadrature rule
5M.B.Giles, ”Multilevel Monte Carlo Path Simulation”, Operations Research, 56, 981-986, 2008
6K.A.Cliffe, et. al. ”Multilevel Monte Carlo Methods and Applications to Elliptic PDEs with Random
Coefficients.” Submitted, to appear in Numerische Mathematik, 2011
Results
The methods are applied to the two-fluid magnetic reconnection problem for
the cases where the electron-to-ion mass ratio, the speed of light and the
initial B-field perturbation are considered stochastic.
LogM of the variance and mean for Pl and
Pl − Pl−1 (velocity at x=0) for the Euler
equations with dispersive source term. The
variance and the mean converge as the
refinement level is increased.
Mean (top) and variance (bottom) of the
reconnected flux for varying speed of light to
Alfven speed ratio ranging from 10 to 20.
Reconnected flux for different plasma models
used in the GEM challenge7 compared to
two-fluid model (red). The error bars are the
standard deviation caused by varying the
c/vA ratio, calculated using the MMC method.
Mean and variance of the reconnected flux
for varying electron-to-ion mass ratio ranging
from 25 to 100.
Mean and variance of the reconnected flux
for varying amplitude of the B-field
perturbation ranging from 8.5% to 11.5% of
the background field.
Computational Cost
The following are actual
computational cost for the case of
the uncertainty in the initial B-field
perturbation.
CPU-hoursMC 6496PCM 4660MMC 4075
ConclusionThe MMC method produced the same accuracy as the standard MC and the
PCM. A 37.3% cost saving was calculated for the MMC method over the MC
and a 14.4% saving over the PCM. MMC allows for an easy and inexpensive
way to determine the error of computational plasma codes.
7 ”Geospace Environmental Modeling (GEM) Magnetic Reconnection Challenge,” Journal of
Geophysical Research, vol. 106, pp. 3715-3719, March 2001.
Computational Plasma Dynamics Lab - Aeronautics and Astronautics Department - University of Washington - Seattle, WA sousae@uw.edu http://www.aa.washington.edu/research/cfdlab
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