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Generalized Fourier Analyses of Semi-Discretizations of the Advection-Diffusion Equation
Sponsored by:ASCI Algorithms - Advanced Spatial Discretization Project
Mark A. Christon, Thomas E. VothComputational Physics R&D (9231)
Mario J. MartinezMultiphase Transport Processes Department (9114)
August 5, 2002
Overview
Background & MotivationPrimer on Fourier Analysis1D and 2D ResultsSummary & Conclusions
Choosing the ‘best’ numerical method can be difficult given the plethora of methods available
Developers embarking on a new code effort are faced with an array of choices for numerical methods
Unstructured vs. structured gridsMesh-full vs. mesh-freeFinite element vs. finite volumeLocal conservation vs. global conservationUse of high-order discretizationsp vs. h-refinement …
Formulation differences, e.g., Galerkin (weighted residual) vs. Taylor-series,can make many aspects of side-by-side comparisons difficult
This work constitutes a first-step in a multi-methods comparison intended to identify strengths and weaknesses
in the context of advection-diffusion processes.
Understanding the behavior of numerical methods in acommon framework is the basis for comparison
Galerkin FEMcos(37 ), sin(37 )o ou v= =
Convected Gaussian cone, from Gresho and Sani, pg. 224, Wiley, 1998
Control Volume FEM
Multi-methods comparisons attempt to identify strengths and weaknesses for a broad range of attributesNumerical performance is a broad term, and includes truncation error, consistency, stability, convergence, etc.
Some desirable attributes:Consistency: discretization recovers the PDE as Stability: , e.g., stable for all and insensitive to roundoffConvergence: numerical solutions approach the PDE solution as
For linear problems, automatic given stability & consistency viaLax’s equivalence theorem
Conservation: 0th-moment should be globally conserved (at a minimum)Non-dispersive: all wavelengths propagate at the ‘true’ advective speedNon-dissipative: there should be no ‘numerical diffusion’Shape-preserving: positivity, monotonicity, lack of spurious oscillationsCompact operators: local ‘stencils’ for the ‘difference’ operatorsComputationally efficient: reasonable memory and CPU requirements
See Baptista, Adams, Gresho, “Benchmarks for the transport equation: the convection-diffusion forum and beyond”, QSACOM, V47, 1995.
t∆1n nu u+ ≤
0, 0x t∆ → ∆ →
0, 0x t∆ → ∆ →
A number of global error measures may be used to assess numerical performance
T exact solution≡
Measure of phase error
Amplitude error orerror in the peak value
Maximum negative value
Measure of spreading
error (maximum local)
error (integral measure of squared error)
error (integral measure of error) ˆ ˆT T d T d− Ω Ω∫ ∫
( )2 2ˆ ˆT T d T d− Ω Ω∫ ∫ˆ ˆmax maxT T T−
max max maxˆ ˆ( )T T T−
max, maxˆnegT T
0 0 0ˆ ˆ( ) ,M M M−
0M xT d T d= Ω Ω∫ ∫2
02
0
( )ˆ ˆ( )
x M T d
x M T d
− Ω
− Ω∫∫
1L
2L
L∞
0.0
0.2
0.4
0.6
0.8
Tem
pera
ture
1.0
0.00 0.50 1.00 1.50 2.00 2.50
x3.00
Phase error
Amplitude error
Spreading
Convergence may not beobserved in all norms onnon-smooth data.
The generalized Fourier analysis places all methods on an equal footing -- regardless of the formulation
Generalized Fourier analysis considers the spectral behavior of semi-discretizations of the advection-diffusion equation
( )
:
: ,
:
1: ,
2
,
,
yx gg
arte
arte art
p
cPhase speed cvv
Group velocity c cDiscrete diffusivity
Artificial diffusivityP
c xP
Short wavelength behaviorGrid anisotropy in terms of
Asymptotic truncation error O xGrid resolution re
αα
α
γ θ
∆=
∆quirements
2T T Tt
α∂+ ∇ = ∇
∂ic
( ) 0MT A T KT+ + =c( )1c lM M Mφ φ= + −
Semi Discretization−:::
wavenumber kdirection
aspect ratioθγ
Inputs Transfer Function
Outputs
cos , sincos , sinx y
u c v ck k k k
ϑ ϑθ θ
= == =
The Fourier analysis places all operators on a ‘regular’ grid configuration with periodic conditions
Fundamental sol’n for A-D:
Substitution into semi-discrete equationyields: For a non-dispersive medium, wavespropagate at the ‘true’ velocity, e.g., in 1-D,Spatial discretization results in dispersivebehavior, i.e., waves propagate at a velocity that is wavelength dependent,
FDM FEM CVFEM
( ) 2, ( ) exp cos sinm nT t A ik m x n y i t k tθ θ ω α = ∆ + ∆ − −
, , , .c k etcω α ω=
Triangular elements require consideration of multiple “regular” grid configurations
c kω=
( )c c k=
The group velocity describes the propagation of ‘wave packets’ in a dispersive medium
Group velocity is defined as:Wave packets consist of short-wavelengthsignals modulating a slow-moving envelope
Energy, , contained in the wave packet moves at the group velocity
Energy may not propagate with the flow, e.g., positive phase with negative group
2∆x oscillations, which may be stationary, appear to move when they modulate a long-wavelength envelope
[ ] ( )tg k kω≡∇ =∇k cv
kκ
exp[ ( (
exp
) )]
[ ( ( ) )
( , )
]ga x v k t
k
T x t
x c k tκκ ι
ι
κ
−
−=
-1.0
-0.5
0.5
1.0
T0.0
0 2 4 6 8 10X
Envelope moves atthe group velocity
( ) 2,T x t dx∫
c
gv
The second-order upwind FDM provides a simple prototype for the generalized Neumann analysis
Complete 1-D nodal equation for
Substitute fundamental solution, calculate phase, group, etc.
Decomposition of the advection operator is automatic
( ) 0MT A T KT+ + =c
2 1
1 12
4 32
2 0
m m m m
m m m
cT T T Tx
T T Txα
− −
− +
+ − +∆
− − + =∆
2( ) expmT t A ikm x i t k tω α = ∆ − −
( ) ( )[ ] ( )[ ] ( ) ( )[ ]
2
2 2 2cos 3 cos 2 4cos
4sin sin
2
22c k x k xi k i
k x c k
x
xx k x
x
ω α
α
+ ∆ − ∆= +
− ∆ +
∆
∆
+ ∆ − ∆∆
Skew-symmetric (non-dissipative)part of advection
Symmetric (dissipative)part of advection
ˆ,Symbol A
,1 1Tskew
TsymA A A A AA = += −
m 1m + 2m +2m − 1m −
x∆
2 2
Non-dimensional parameters characterizephase and group speed, discrete and artificial diffusivity
( )[ ]
( ) ( )[ ]
2 2
2 2
1 2 2cos
21 1 3 cos 2 4cos
art
artarte
k xk x
k x k xc xP k x
α α ααα
α
= +
= − ∆∆
= = + ∆ − ∆∆ ∆
Imaginary part of symbol, , yields the phase speed
Real part of symbol, , yields the discrete and artificial diffusivity
Characterizing the discretization:
( )( )
ˆRe( ( )) 0 ,ˆRe ( ) 0 ,
ˆRe ( ) 0 ,
A k for all k discretization is neutrally dissipative
A k for some k discretization is dissipative
A k for some k discretization is unstable
=
<
>
ˆIm( )A( ) ( )[ ]1 4sin sin 2
2c k x k xc k x= ∆ − ∆
∆ˆRe( )A
Necessary & Sufficient:is skew-symmetricis symmetric, e.g.,
Galerkin FEM
( )A cM
Phase Speed Group Speed Artificial Diffusivity0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
~c/c
0.0 0.2 0.4 0.6 0.8
2 ∆x/λ
1.0
-0.50-0.250.000.250.500.751.001.25
T
1.50
0.0 0.2 0.4 0.6 0.8x
1.0
t = 7.5 (u ∆t/∆x)
t = 7.5 (u ∆t/∆x)
~ 3∆x2∆x
-0.50-0.250.000.250.500.751.001.251.50
T
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1/Peart
0.0 0.2 0.4 0.6 0.8 1.0
2∆x/λ
-3.0
-2.0
-1.0
0.0
1.0
~vgx/c
2.0
0.0 0.2 0.4 0.6 0.8 1.0
2 ∆x/λ
t = 3.75 (u ∆t/∆x)
~ 3∆x2∆x
-0.50-0.250.000.250.500.751.001.25
T
1.50
0.0 0.2 0.4 0.6 0.8 1.0
t = 3.75 (u ∆t/∆x)
-0.50-0.250.000.250.500.751.001.251.50
T
0.0 0.2 0.4 0.6 0.8 1.0x
( )1 2 TskewA A A= −
2k x xπ λ∆ = ∆
SOUSkew-Symmetric
Advection
SOUFull Advection
Operator
There is a direct relationship between order of accuracy and the ‘flatness’ of phase (and diffusivity) near k∆x = 0
10-5
10-4
10-3
10-2
10-1
100
10-1 100
Truncation error:
Phase error:
TE and phase (and diffusivity) error yield order accuracy, p:
( ), 1,TE m nm n
dTc T
dt−= −M A
( ) TOHxkCcc p ..1~
+∆=− ι
TOHxTxC p
pp ..TE 1
1+
∂∂
∆= +
+
( ) 0→∆∆= xasxOTE p
( )( ) 01~
→∆∆=− xkasxkOcc p
1
4
121/~ −cc
/ ( 2 / )k x xπ λ∆ = ∆
CD-FDM
FEM-Mc
0.0
0.2
0.4
0.6
0.8
1.0
c~/c
0.0 0.2 0.4 0.6 0.8 1.0
CD-FDM
FEM-Mc
Analysis of non-linear methods yields an ‘operating range’ for phase speed and artificial diffusivity
Godunov-type central scheme (Kurganov & Tadmor, 2000) using minmod slope limiter
( ) ( )( )
1
11
min 0,max 1, ,
m mm
m
m m m
m mm
m m
T TTx x
T TT T
φ
φ θ θ
θ
+
−
+
−∂ = ∂ ∆
=
−=
−
FOU/CD
FOU1 0, 1m mφ φ− = =
1 1, 0m mφ φ− = =
1 1, 1m mφ φ− = =
1 0, 0m mφ φ− = =
m-2 m-1 m m+10.00
0.20
0.40
0.60
0.80
1.00
1/Peart
0.0 0.2 0.4 0.6 0.8 1.0
2∆x/λ
0.0
0.5
1.0
1.5
c~/c
0.0 0.2 0.4 0.6 0.8 1.0
1 0, 0m mφ φ− = =
1 0, 1m mφ φ− = =
1 1, 0m mφ φ− = =
1 1, 1m mφ φ− = =
1 0, 0m mφ φ− = =
1 0, 1m mφ φ− = =
1 1, 0m mφ φ− = =
A broad cross-section of methods have been considered in this study
Least squares reconstruction (LSR(0), LSR(-1))
Second-order central differences (Centered FDM)
Streamline-upwind control-volume finite element (CVFEM-SUCV)
Control-volume finite element (CVFEM)
Streamline-upwind Petrov-Galerkin (FEM-SUPG)
Galerkin finite element (FEM)
QUICK (Quadratic Upwind Interpolation with Correct Kinematics)
Fromm’s method (semi-discrete version)
Third-order upwind (TOU)
Second-order upwind (SOU)
First-order upwind (FOU)
.
1 15
.1 2
opt
SUPG Opt
SUCV Opt
ββ
==
FEM with consistent mass delivers the best overall phase speed with superconvergent asymptotic behavior
1%error
5%error
2.695.71
6.24
4.392.883.933.966.835.4615.8
11.4
12.8Ο(∆x2)CVFEM-SUCV (βopt)
6.78Ο(∆x4)FEM-SUPG (β= ½)
5.61Ο(∆x4)FEM-Mc
11.8Ο(∆x2)CVFEM-SUCV (β= ½)
13.1Ο(∆x2)CVFEM-Mc
4.76Ο(∆x6)FEM-SUPG (βopt)
17.4Ο(∆x2)Fromm’s13.5Ο(∆x2)QUICK8.35Ο(∆x4)TOU36.2Ο(∆x2)SOU
25.6Ο(∆x2)FOUFEM-Ml
CVFEM-Ml
λ/∆x forTEMethod
CVFEM-Mc
FEM-Mc
FEM – SUPG (βopt)
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
CVFEM – SUCV (β = ½) c / c
Centered FDM
QUICK
TOU
Fromm's
SOU
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
c / c
2∆x/λ
FEM with consistent mass delivers the best overall group speed with superconvergent asymptotic behavior
1%error
5%error
3.139.88
10.4
6.703.755.7011.811.28.2927.7
19.7
22.2Ο(∆x2)CVFEM-SUCV (βopt)
10.3Ο(∆x4)FEM-SUPG (β= ½)
8.31Ο(∆x4)FEM-Mc
21.3Ο(∆x2)CVFEM-SUCV (β= ½)
22.5Ο(∆x2)CVFEM-Mc
4.62Ο(∆x6)FEM-SUPG (βopt)
30.8Ο(∆x2)Fromm’s22.9Ο(∆x2)QUICK12.6Ο(∆x4)TOU62.7Ο(∆x2)SOU
44.4Ο(∆x2)FOUFEM-Ml
CVFEM-Ml
λ/∆x for ARMethod
-3
-2
-1
0
1
2
Centered FDM
QUICK
TOU
Fromm's
SOU
gv / c
gv / c
-6
-5
-4
-3
-2
-1
0
1
2
0.0 0.2 0.4 0.6 0.8 1.0
FEM – SUPG (βopt)
CVFEM – SUCV (β = ½)
FEM-Mc
CVFEM-Mc
2∆x/λ
Two-Dimensional spatial discretization introduces wavelength AND direction dependent behavior
θ
∆x
∆y
2∆x/λ1.00.80.60.40.20.0
0.000°
90°
270°
1.50180°
22.5, 67.5o
45o
0, 90o
0.0 0.2 0.4 0.6 0.8
2 ∆x/λ
1.00.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
~c/cθ
2∆x/λcc /~
cc /~
0.00.2
0.40.6
0.81.0
0.010.0
20.030.0
40.050.0
60.070.0
80.090.0
0.00.2
0.4
0.6
0.8
θ
1.0
1.2
2 ∆x/λ
~c/c
1.4
SOU
Grid Definitions
Integrated anisotropy and error metrics used to allow ‘objective’ comparison of two-dimensional results
( ) ( )( ) dkdkckck∫ ∫ −=
θθθσ 2~,~Anisotropy metric:
( )( ) dkdckck∫ ∫ −=
θθθε 2,~Error metric:
2∆x/λ1.00.80.60.40.20.0
0.000°
90°
180°1.50
270°
0.00.20.40.60.81.0
2∆x/λ
0.000°
90°
270°
1.25180°
large anisotropy (1.7e-1)large error (3.3e-1)
small anisotropy (8.8e-2)large error (2.2e-1)
FOU LSR(-1)
cc /~ cc /~
2∆x/λ1.00.80.60.40.20.0
0.000°
90°
270°
1.25180°
small anisotropy (7.3e-2)small error (7.1e-2)
CVFEM-SUCV (β = ½)
cc /~
Integrated anisotropy and phase error metrics suggest FEM-SUPG and CVFEM-SUCV schemes are best
7.1e-27.3e-2CVFEM-SUCV (β= ½)
1.3e-17.8e-2FEM-SUPG (β= ½)
2.1e-19.1e-2LSR(0)2.2e-18.8e-2LSR(-1)
1.5e-11.1e-1CVFEM-SUCV (βopt)2.1e-11.3e-1CVFEM-Mc
7.3e-27.9e-2FEM-SUPG (βopt)1.4e-11.1e-1FEM-Mc
1.8e-11.4e-1Fromm’s2.4e-11.5e-1QUICK2.2e-11.4e-1TOU2.2e-11.4e-1SOU3.3e-11.7e-1FOU
Error, eAnisotropy, sMethod0.00.20.40.60.81.0
2∆x/λ
0.000°
90°
270°
1.25180°
2∆x/λ1.00.80.60.40.20.0
0.000°
90°
270°
1.25180°
CVFEM-SUCV (β = ½)
FOU
cc /~
cc /~
Unlike the physical problem, the discrete problem introduces a diffusivity that varies by wavelength
Consider 1-D transient diffusion in a bar:
Analytical sol’n is:
Numerical sol’n. is similar, but with wavelength dependent diffusivity:
( ) ( )[ ] [ ]tkxkBtxT nn
nn2
1expsin, α−=∑
∞
=
T = 0 T = 0
x
0.0 0.2 0.4 0.6 0.8 1.0x-0.1
0.0
0.1
0.2
0.3
0.4
0.5
T
initial
0.20.0 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
αα~
λ/2 x∆
( ) ( )[ ] [ ]tkxkBtT nnn
inni2
1
~expsin α−=∑∞
=
SUPG and SUCV formulations with consistent mass and β = βopt yield best discrete diffusivity results
1%error
5%error
12.32.30
5.4511.04.208.19
8.03
9.52Ο(∆x2)CVFEM-SUCV (βopt)28.4Ο(∆x2)CVFEM-SUCV (β=½)
8.37Ο(∆x2)FEM-SUPG (βopt)25.5Ο(∆x2)FEM-SUPG (β= ½)
18.1Ο(∆x2)CD-FDMFEM- ML
CVFEM-ML
12.8Ο(∆x2)CVFEM-Mc
18.2Ο(∆x2)FEM-Mc
λ/∆x forTEMethodCVFEM - M c
FEM - M c
FEM/CVFEM - M l
0.00
0.25
0.50
0.75
1.00
1.25
1.50
α~/α
FEM - SUPG β=1/2
FEM - SUPG βopt
CVFEM - SUCV βopt
CVFEM - SUCV β=1/2
0.00
0.25
0.50
0.75
1.00
1.25
1.50
α~/α
0.0 0.2 0.4 0.6 0.8 1.0
2∆x /λ
Note: βopt is NOT the optimalchoice for SUCV phase
FEM-Mc and CVFEM-Mc demonstrate best anisotropy and error respectively for discrete diffusivity
0.00.20.40.60.81.0
2∆x/λ
0.00°
90°
180°1.5
270°
3.2e-17.3e-2CVFEM-SUCV (β=½)
3.1e-11.7e-1FEM-SUPG (β=½)
1.1e-15.1e-2CVFEM-SUCV (βopt)
5.3e-25.3e-2CVFEM-Mc
9.8e-28.6e-2FEM-SUPG (βopt)1.8e-12.8e-2FEM-Mc
1.8e-11.7e-1CD-FDMError, eAnisotropy, sMethod
CVFEM-SUCV (β = ½)
FEM-Mc2∆x/λ
1.00.80.60.40.20.0
0.000°
90°
270°
1.25180°
αα /~
αα /~
Artificial diffusivity should increase with wave number suggesting poor behavior for FOU
FOU artificial diffusivity acts at ALL wavelengths
CVFEM - SUPG βopt
FEM - SUPG β = 1/2
CVFEM - SUPG β=1/2
FEM - SUPG βopt
0.0 0.2 0.4 0.6 0.8 1.0
2∆x /λ0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1/Peart
Second-Order Upwind (FOU)
Third-Order Upwind (FOU)
QUICK
Fromm's Method
First-Order Upwind (FOU)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1/Peart
CVFEM-Mc
FEM-Mc
FEM – SUPG (βopt)
0.0
0.2
0.4
0.6
0.8
1.0
CVFEM – SUCV (β = ½) c / c
-6
-5
-4
-3
-2
-1
0
1
2
0.0 0.2 0.4 0.6 0.8 1.0
FEM – SUPG (βopt)
CVFEM – SUCV (β = ½)
FEM-Mc
CVFEM-Mc
2∆x/λ
/gv c
Quadratic energy damping characteristics suggest that the FEM and CVFEM SU variants are good choices
t tt x / c t
tt
QT τ τ
τ
∆T MT T MT
T MT= +∆ =
=
−=
First-Order Upwind (FOU)
Second-Order Upwind (SOU)Fromm's Method
Third-Order Upwind (TOU)
QUICK
-1.0
-0.8
-0.4
-0.2
0.0
∆QT
-0.6
0.0 0.2 0.4 0.6 0.8 1.02∆x /λ
FEM - SUPG βopt
FEM - SUPG β = 1/2
CVFEM - SUCV β = 1/2
CVFEM - SUCV βopt
0.0 0.2 0.4 0.6 0.8 1.02∆x /λ
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
∆QT
Finite Difference schemes FEM-SUPG and CVFEM-SUCV schemes
‘Constant’ mode isn’t damped
Signal not damped in ∆x/c time scale
Summary and Conclusions
In the discrete world, Waves don’t propagate at the advective velocity, and don’t always propagate in the direction of the wave vectorInformation doesn’t diffuse at the continuum rateGrid bias will be present … Fourier analysis can quantify these errors
Some clear losers and not so obvious winnersClear ‘losers’ are FOU and SOU
SOU provides the WORST phase and group, requiring the most resolution for a fixed accuracy levelFOU, lumped mass FEM and CVFEM also perform relatively poorly
Choice of a winner is more difficultUpwind (SU) variants of FEM and CVFEM perform well with careful choice of stabilization parameterFEM variants w. consistent mass yield superconvergent phase and group with minimum mesh resolution requirements
TOU also exhibits superconvergent behaviorAnalysis of non-linear methods is possible, but results are not as ‘sharp’ as for linear methods
A number of diagnostic metrics can be used toassess the numerical performance
For with no-flux or periodic BC’s0, 0T T witht
∂+ ∇ = ∇ =
∂i ic c
Ability to preserve gradients
Ability to preserve peaks
Quadratic (“energy”) conservation
Ability to preserve curvature
Global conservation T dΩ
Ω∫2T d
ΩΩ∫
4T dΩ
Ω∫
T T dΩ∇ ∇ Ω∫ i
22T dΩ
∇ Ω∫
Phase speed is the projection of the fluid velocity in the wave direction -- the “apparent velocity”
Scalar advection:
Use a fundamental sol’n
Solve for the circular frequency
and phase speed
For a non-dispersive medium, wavespropagate at the “true” velocity, e.g., in 1-D,Spatial discretization results in dispersivebehavior, i.e., waves don’t propagate at the “true” velocity and
Waves propagate at a velocity that is wavelength dependent
[ , ]tx yk k=k[ , ]t u v=c
troughcrest
c
k
x
y0T T Tu v
t x y∂ ∂ ∂+ + =∂ ∂ ∂
tx yk u k vω = + = k c
( , ) exp[ ( ) ]x yT t A k x k y tι ιω= + −x
c u=
/tc ≡ k c k
c u≠
(Adapted from Gresho’s notes, Taiwan course, 1989)
Two-dimensional analysis reveals angulardependence (grid-bias) in phase, group, and diffusivities
Group Velocity (x-dir)0.0
0.20.4
0.60.8
1.0
0.010.0
20.030.0
40.050.0
60.070.0
80.090.0
-6.0-5.0-4.0-3.0-2.0-1.0
θ
0.01.0
2∆x/λ
2.0
v~gx/c
Phase Speed0.0
0.20.4
0.60.8
1.0
0.010.0
20.030.0
40.050.0
60.070.0
80.090.0
0.00.20.40.60.8
θ
1.0
1.2
2 ∆x/λ
~c/c
1.4
Discrete Diffusivity
0.00.2
0.40.6
0.81.0
0.0
18.036.0
54.072.0
90.0
0.0
0.2
0.4
0.6
0.8
θ
1.0
2∆x/λ
∼α/α
Artificial Diffusivity
0.00.2
0.40.6
0.81.0
0.010.0
20.030.0
40.050.0
60.070.0
80.090.0
0.0
0.2
0.4
0.6
θ
0.8
1.0
2∆x/λ
1.2
1/Parte
Results Outline
Phase and Group SpeedOne-Dimensional phase and group resultsTwo-Dimensional phase results
Discrete DiffusivityOne-Dimensional resultsTwo-Dimensional results
Artificial DiffusivityOne-Dimensional resultsTwo-Dimensional results
Grid aspect ratio modifies the anisotropic phase behavior of the two-dimensional discretizations
γ = 1/2Anisotropy, σMethod
4.9e-27.3e-2CVFEM-SUCV (β= ½)
6.5e-27.8e-2FEM-SUPG (β= ½)
1.2e-19.1e-2LSR(0)
9.4e-21.1e-1CVFEM-SUCV (βopt)1.3e-11.3e-1CVFEM-Mc
5.6e-27.9e-2FEM-SUPG (βopt)9.4e-21.1e-1FEM-Mc
1.2e-11.4e-1Fromm’s1.5e-11.5e-1QUICK1.4e-11.4e-1TOU1.2e-11.4e-1SOU1.8e-11.7e-1FOU
γ = 1
FEM
-SU
PG (β o
pt)
γ = 1
γ = 1/2
2∆x/λ1.00.80.60.40.20.0
0.000°
90°
270°
1.25180°
0.00.20.40.60.81.0
2∆x/λ
0.000°
90°
270°
1.25180°
cc /~
cc /~
Artificial diffusivity should increase with wave number suggesting poor behavior for FOU
Ο(∆x2)CVFEM-SUPG (βopt)
Ο(∆x2)FEM-SUPG (βopt)Ο(∆x2)Fromm’s
Ο(∆x2)CVFEM-SUPG (β=½)
Ο(∆x2)FEM-SUPG (β=½)
Ο(∆x2)QUICKΟ(∆x2)TOUΟ(∆x2)SOUΟ(1)FOUT.E.Method
Second-Order Upwind (FOU)
Third-Order Upwind (FOU)
QUICK
Fromm's Method
First-Order Upwind (FOU)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1/Peart
CVFEM - SUPG βopt
FEM - SUPG β = 1/2
CVFEM - SUPG β=1/2
FEM - SUPG βopt
0.0 0.2 0.4 0.6 0.8 1.0
2∆x /λ0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1/Peart
FOU artificial diffusivityacts at ALL wavelengths
The spectral dependence of the artificial diffusivity does not completely explain its damping effects
Given the fundamental solution:
Damping of T depends on αart and k:
Similar dependency for QT=TtMT:
( ) )exp()(, 20 tkTtT artα−= xx
( ) )exp()(, 20
2 tkTkttT artart αα −−=∂∂ xx
( ) ( )0
222
2exp QTtkk
tQT artart αα −−
=∂
∂
0.0
0.5
1.0
-0.5
-1.0
T
-1.0
-0.5
0.0
0.5
1.0
0.0 0.2 0.4 0.6 0.8 1.0x0.0 0.2 0.4 0.6 0.8 1.0-1.0
-0.5
0.0
0.5
1.0
T
t = .01/αart
t = 0
k = 2π
k = 4π
0.0
0.2
0.4
0.6
0.8
1.0
0.000 0.005 0.010 0.015 0.020 0.025 0.0300.000 0.005 0.010 0.015 0.020 0.025 0.030
tαart
0.0
0.2
0.4
0.6
0.8
1.0
QT(
t)/Q
T(t=
0)
Least-Squares Gradient Reconstruction schemes minimize angular dependence of artificial diffusivity.
1.6e-13.4e-1CVFEM-SUCV (βopt)2.4e-14.5e-1FEM-SUPG (β=½)1.7e-13.8e-1FEM-SUPG (βopt)
2.3e-14.1e-1CVFEM-SUCV (β=½)
3.1e-14.5e-2LSR(0)6.1e-14.8e-2LSR(-1)
2.6e-12.1e-1Fromm’s
2.5e-22.1e-1QUICK1.7e-12.1e-1TOU5.1e-12.1e-1SOU6.9e-17.8e-2FOU
DiffusivityAnisotropyMethod
FEM-SUPG (β = ½)
LSR(0)
0.00.20.40.60.81.0
2∆x/λ
0.00°
90°
180°0.6
270°
2∆x/λ1.00.80.60.40.20.0
0.000°
90°
270°
1.50180°
artPe/1
artPe/1