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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