Generalized Fourier Analyses of Semi-Discretizations of the Advection-Diffusion Equation

Transcription

1 Generalized Fourier Analyses of Semi-Discretizations of the Advection-Diffusion Equation Sponsored by: ASCI Algorithms - Advanced Spatial Discretization Project Mark A. Christon, homas E. Voth Computational Physics R&D (931) Mario J. Martinez Multiphase ransport Processes Department (9114) August 5, 00

2 Overview Background & Motivation Primer on Fourier Analysis 1D and D Results Summary & Conclusions

3 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 grids Mesh-full vs. mesh-free Finite element vs. finite volume Local conservation vs. global conservation Use of high-order discretizations p vs. h-refinement Formulation differences, e.g., Galerkin (weighted residual) vs. aylor-series, can make many aspects of side-by-side comparisons difficult his work constitutes a first-step in a multi-methods comparison intended to identify strengths and weaknesses in the context of advection-diffusion processes.

4 Understanding the behavior of numerical methods in a common framework is the basis for comparison Galerkin FEM Control Volume FEM o o u = cos(37 ), v= sin(37 ) Convected Gaussian cone, from Gresho and Sani, pg. 4, Wiley, 1998

5 Multi-methods comparisons attempt to identify strengths and weaknesses for a broad range of attributes Numerical performance is a broad term, and includes truncation error, consistency, stability, convergence, etc. Some desirable attributes: Consistency: discretization recovers the PDE as x 0, t 0 Stability: n+ 1 n u u, e.g., stable for all t and insensitive to roundoff Convergence: numerical solutions approach the PDE solution as x For linear problems, automatic given stability & consistency via Lax s equivalence theorem Conservation: 0 th -moment should be globally conserved (at a minimum) Non-dispersive: all wavelengths propagate at the true advective speed Non-dissipative: there should be no numerical diffusion Shape-preserving: positivity, monotonicity, lack of spurious oscillations Compact operators: local stencils for the difference operators Computationally efficient: reasonable memory and CPU requirements See Baptista, Adams, Gresho, Benchmarks for the transport equation: the convection-diffusion forum and beyond, QSACOM, V47, , t 0

6 A number of global error measures may be used to assess numerical performance 1 L error (integral measure of error) L error (integral measure of squared error) L error (maximum local) ˆ dω ˆ dω ( ˆ) ˆ dω dω max ˆ max ˆ Convergence may not be observed in all norms on non-smooth data. ˆ exact solution Amplitude error or error in the peak value Maximum negative value Measure of phase error Measure of spreading ( ˆ ) ˆ max max max max, neg ˆ max ( M Mˆ ) Mˆ, M = x dω dω ( x M ) dω ( x Mˆ ) d ˆ Ω emperature Amplitude error Spreading Phase error x 3.00

7 he 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 Inputs + ci = α t wavenumber : k direction : θ aspect ratio : γ ransfer Function Semi Discretization M + A() c + K = 0 c ( 1 φ ) M = φm + M u = ccos ϑ, v= csinϑ k = kcos θ, k = ksinθ x y l Outputs Phase speed : c c v g v y : x g Group velocity, c c Discrete diffusivity : α α Artificial diffusivity : 1 art, P e art P c x e = α art Short wavelength behavior Grid anisotropy in terms of γθ, Asymptotic truncation error, O Grid resolution requirements p ( x )

8 he Fourier analysis places all operators on a regular grid configuration with periodic conditions FDM FEM CVFEM Fundamental sol n for A-D: mn, () t = Aexp ik( m xcosθ + n ysinθ) i ωt k αt Substitution into semi-discrete equation yields: ω, α, c = ω k, etc. For a non-dispersive medium, waves propagate at the true velocity, e.g., in 1-D, c=ω k Spatial discretization results in dispersive behavior, i.e., waves propagate at a velocity that is wavelength dependent, c ( ) = c k riangular elements require consideration of multiple regular grid configurations

9 he group velocity describes the propagation of wave packets in a dispersive medium t Group velocity is defined as: vg k[ kc] ( = kω) Wave packets consist of short-wavelength signals modulating a slow-moving envelope κ ( x, t) = aκ exp[ ικ ( x vg ( k))] t ( ) exp[ ιk( x c( k) t) ] Energy, x, t dx, contained in the wave packet moves at the group velocity Energy may not propagate with the flow, e.g., positive phase with negative group c v g -0.5 κ k x oscillations, which may be stationary, appear to move when they modulate a long-wavelength envelope 0.5 Envelope moves at the group velocity X

10 he second-order upwind FDM provides a simple prototype for the generalized Neumann analysis Complete 1-D nodal equation for M + A() c + K = 0 c m + { m 4m 1+ 3m} x α { 1 1} 0 m m + m+ = x Substitute fundamental solution, calculate phase, group, etc. m () t = Aexp ikm x i ωt k αt { c [ ( ) ( )]} 4sin k x sin k x i ω + αk = i + x α x m m 1 c x [ cos( k x) ] + 3+ cos( k x) 4cos( k x) [ ] Decomposition of the advection operator is automatic 1 1 Askew = A A, A sym = A+ A x m m + 1 m + Skew-symmetric (non-dissipative) part of advection Symbol, Aˆ Symmetric (dissipative) part of advection

11 Non-dimensional parameters characterize phase and group speed, discrete and artificial diffusivity Imaginary part of symbol, Im( Aˆ ), yields the phase speed c 1 = [ 4sin( k x) sin( k x) ] c k x Real part of symbol, Re( Aˆ ), yields the discrete and artificial diffusivity α = α + α art α 1 = [ cos( k x) ] α k x 1 α art 1 = = + art c x P k x e [ 3 cos( k x) 4cos( k x) ] Characterizing the discretization: Re( Aˆ ( k)) = 0 for all k, discretization is neutrally dissipative ( ˆ ) ( ˆ ) Re A( k) < 0 for some k, discretization is dissipative Re A( k) > 0 for some k, discretization is unstable Necessary & Sufficient: A() c is skew-symmetric M is symmetric, e.g., Galerkin FEM

12 ~c/c Phase Speed Group Speed Artificial Diffusivity x/λ k x π = x λ SOU Skew-Symmetric Advection A 1( ) skew = A A SOU Full Advection Operator ~v gx /c t = 3.75 (u t/ x) x x/λ ~ 3 x t = 3.75 (u t/ x) x 1/Pe art t = 7.5 (u t/ x) x t = 7.5 (u t/ x) ~ 3 x x

13 here is a direct relationship between order of accuracy and the flatness of phase (and diffusivity) near k x = 0 runcation error: dmn, 1 E = M A( c ) mn, dt p+ 1 p E = C x H. O. p 1 x + + Phase error: c~ p 1 = C( k ι x) + H. O. c E and phase (and diffusivity) error yield order accuracy, p: p ( x ) as 0 E = O x c~ p 1 = O ( k x) c as ( ) k x 0 c~ /c c ~ / c CD-FDM 4 CD-FDM FEM-M c k x/ π ( = x/ λ) FEM-M c

14 Analysis of non-linear methods yields an operating range for phase speed and artificial diffusivity Godunov-type central scheme (Kurganov & admor, 000) using 1.5 minmod slope limiter φ m+ 1 m = m x m x ( ) = ( ( )) φ θ min 0, max 1, θ, θ m m m m = m m+ 1 m 1 m c~ /c 0.5 FOU/CD φm 1 = 0, φm = 0 φm 1 = 0, φm = 1 φm 1 = 1, φm = 0 φm 1 = 1, φm = 1 φm 1 = 0, φm = 0 0 φm 1 = 0, φm = 1 φm 1 = 1, φm = 0 φm 1 = 1, φm = 1 1/Pe art FOU φm 1 = 0, φm = 0 φm 1 = 0, φm = 1 φm 1 = 1, φm = 0 m- m-1 m m+1 0

15 A broad cross-section of methods have been considered in this study First-order upwind (FOU) Second-order upwind (SOU) Second-order central differences (Centered FDM) hird-order upwind (OU) Fromm s method (semi-discrete version) QUICK (Quadratic Upwind Interpolation with Correct Kinematics) Least squares reconstruction (LSR(0), LSR(-1)) Galerkin finite element (FEM) Streamline-upwind Petrov-Galerkin (FEM-SUPG) Control-volume finite element (CVFEM) Streamline-upwind control-volume finite element (CVFEM-SUCV) β SUPG Opt. β = 1 15 = opt SUCV Opt. 1

16 FEM with consistent mass delivers the best overall phase speed with superconvergent asymptotic behavior Fromm's Method E λ/ x for 5% 1% error error c/c c/c Centered FDM QUICK OU CVFEM-M c FEM-M c CVFEM SUCV (β = ½) FEM SUPG (β opt ) SOU FOU FEM-M l CVFEM-M l SOU OU QUICK Fromm s FEM-M c FEM-SUPG (β opt ) FEM-SUPG (β= ½) CVFEM-M c CVFEM-SUCV (β opt ) Ο( x 4 ) Ο( x 4 ) Ο( x 6 ) Ο( x 4 ) CVFEM-SUCV (β= ½)

17 1 0 v /c g g v /c FEM with consistent mass delivers the best overall group speed with superconvergent asymptotic behavior QUICK OU Centered FDM CVFEM-M c FEM-M c FEM SUPG (β opt ) CVFEM SUCV (β = ½) SOU Fromm's -6 FOU FEM-M l CVFEM-M l SOU OU QUICK Fromm s FEM-M c FEM-SUPG (β opt ) FEM-SUPG (β= ½) CVFEM-M c Method CVFEM-SUCV (β opt ) CVFEM-SUCV (β= ½) AR Ο( x 4 ) Ο( x 4 ) Ο( x 6 ) Ο( x 4 ) 5% error λ/ x for 1% error

18 180 wo-dimensional spatial discretization introduces wavelength AND direction dependent behavior c ~ / θ c c ~ / ~c/c c SOU , 90 o.5, 67.5 o x/λ 45 o y x Grid Definitions θ θ x/λ ~c/c

19 Integrated anisotropy and error metrics used to allow objective comparison of two-dimensional results Anisotropy metric: Error metric: σ ε k ( c~ ( θ, k) c~ ( k) ) dθ dk = k θ ( c ( θ, k) c) dθ dk = θ ~ c ~ / c c ~ / c c ~ / c CVFEM-SUCV (β = ½) small anisotropy (7.3e-) small error (7.1e-) 70 FOU large anisotropy (1.7e-1) large error (3.3e-1) 70 LSR(-1) small anisotropy (8.8e-) large error (.e-1)

20 Integrated anisotropy and phase error metrics suggest FEM-SUPG and CVFEM-SUCV schemes are best c ~ / c FOU SOU OU QUICK Method Anisotropy, s 1.7e-1 1.4e-1 1.4e-1 1.5e-1 Error, e 3.3e-1.e-1.e-1.4e-1 Fromm s 1.4e-1 1.8e FOU c ~ / c LSR(0) LSR(-1) FEM-M c FEM-SUPG (β opt ) FEM-SUPG (β= ½) CVFEM-M c CVFEM-SUCV (β opt ) 9.1e- 8.8e- 1.1e-1 7.9e- 7.8e- 1.3e-1 1.1e-1.1e-1.e-1 1.4e-1 7.3e- 1.3e-1.1e-1 1.5e-1 70 CVFEM-SUCV (β= ½) CVFEM-SUCV (β = ½) 7.3e- 7.1e-

21 Unlike the physical problem, the discrete problem introduces a diffusivity that varies by wavelength Consider 1-D transient diffusion in a bar: 0.5 initial = 0 = 0 Analytical sol n is: x [ ] [ exp k t] ( x t) B sin( k x), = n n α = n 1 Numerical sol n. is similar, but with wavelength dependent diffusivity: i [ ] [ exp ~ α k t] () t B sin( k x ) = = n 1 n n i n n n α ~ x 1.5 α 0.5 x / λ

22 SUPG and SUCV formulations with consistent mass and β = β opt yield best discrete diffusivity results 1.50 α~ / α FEM - M c CVFEM - M c FEM/CVFEM - M l Method CD-FDM FEM- M L CVFEM-M L E λ/ x for 5% 1% error error FEM-M c FEM - SUPG β opt CVFEM - SUCV β opt FEM-SUPG (β opt ) FEM-SUPG (β= ½) CVFEM-M c α ~ /α 0 CVFEM-SUCV (β opt ) CVFEM-SUCV (β=½) CVFEM - SUCV β=1/ FEM - SUPG β=1/ Note: β opt is NO the optimal choice for SUCV phase 0 x /λ

23 FEM-M c and CVFEM-M c demonstrate best anisotropy and error respectively for discrete diffusivity ~ α /α Method CD-FDM FEM-M c Anisotropy, s 1.7e-1.8e- Error, e 1.8e-1 1.8e-1 FEM-SUPG (β opt ) 8.6e- 9.8e- FEM-SUPG (β=½) 1.7e-1 3.1e FEM-M c CVFEM-M c 5.3e- 5.3e- ~ α /α CVFEM-SUCV (β opt ) CVFEM-SUCV (β=½) 5.1e- 7.3e- 1.1e-1 3.e CVFEM-SUCV (β = ½)

24 Artificial diffusivity should increase with wave number suggesting poor behavior for FOU /Pe art FOU artificial diffusivity acts at ALL wavelengths Second-Order Upwind (FOU) First-Order Upwind (FOU) Fromm's Method c/c CVFEM-M c FEM-M c CVFEM SUCV (β = ½) FEM SUPG (β opt ) hird-order Upwind (FOU) QUICK 1.4 1/Pe art 1. FEM - SUPG β = 1/ CVFEM - SUPG β=1/ FEM - SUPG β opt CVFEM - SUPG β opt x /λ v g / c CVFEM-M c FEM-M c FEM SUPG (β opt ) CVFEM SUCV (β = ½) -6

25 Quadratic energy damping characteristics suggest that the FEM and CVFEM SU variants are good choices Q = Constant mode isn t damped { t t M M } t= τ + x/c t= τ t M t= τ - QUICK - FEM - SUPG β = 1/ hird-order Upwind (OU) - - FEM - SUPG β opt Q - Fromm's Method Second-Order Upwind (SOU) Q - CVFEM - SUCV β = 1/ CVFEM - SUCV β opt - - First-Order Upwind (FOU) - x /λ Finite Difference schemes - x /λ FEM-SUPG and CVFEM-SUCV schemes Signal not damped in x/c time scale

26 In the discrete world, Summary and Conclusions Waves don t propagate at the advective velocity, and don t always propagate in the direction of the wave vector Information doesn t diffuse at the continuum rate Grid bias will be present Fourier analysis can quantify these errors Some clear losers and not so obvious winners Clear losers are FOU and SOU SOU provides the WORS phase and group, requiring the most resolution for a fixed accuracy level FOU, lumped mass FEM and CVFEM also perform relatively poorly Choice of a winner is more difficult Upwind (SU) variants of FEM and CVFEM perform well with careful choice of stabilization parameter FEM variants w. consistent mass yield superconvergent phase and group with minimum mesh resolution requirements OU also exhibits superconvergent behavior Analysis of non-linear methods is possible, but results are not as sharp as for linear methods

27 A number of diagnostic metrics can be used to assess the numerical performance For + ci = 0, with ic = 0 with no-flux or periodic BC s t Global conservation Quadratic ( energy ) conservation Ability to preserve peaks Ability to preserve gradients Ability to preserve curvature Ω Ω Ω Ω Ω dω 4 dω dω i dω { } d Ω

28 Phase speed is the projection of the fluid velocity in the wave direction -- the apparent velocity Scalar advection: + u + v = 0 t c = [ uv, ] t x y Use a fundamental sol n y ( x, t) = Aexp[ ι( kxx+ kyy) ιωt] Solve for the circular frequency k t ω = ku x + kv y =kc and phase speed t c kc/ k For a non-dispersive medium, waves propagate at the true velocity, e.g., in 1-D, c = u Spatial discretization results in dispersive behavior, i.e., waves don t propagate at the true velocity and c u Waves propagate at a velocity that is wavelength dependent (Adapted from Gresho s notes, aiwan course, 1989) k t crest trough = [ k, k ] c x y x

29 wo-dimensional analysis reveals angular dependence (grid-bias) in phase, group, and diffusivities θ 1 Phase Speed x/λ ~c/c θ 1 Group Velocity (x-dir) v~ gx /c θ 18.0 Discrete Diffusivity α/α θ 1 Artificial Diffusivity 1/P art e

30 Results Outline Phase and Group Speed One-Dimensional phase and group results wo-dimensional phase results Discrete Diffusivity One-Dimensional results wo-dimensional results Artificial Diffusivity One-Dimensional results wo-dimensional results

31 Grid aspect ratio modifies the anisotropic phase behavior of the two-dimensional discretizations FEM-SUPG (β opt ) c ~ / γ = 1 c ~ / c c Method FOU SOU OU QUICK Fromm s LSR(0) FEM-M c FEM-SUPG (β opt ) FEM-SUPG (β= ½) CVFEM-M c CVFEM-SUCV (β opt ) Anisotropy, σ γ = 1 γ = 1/ 1.7e-1 1.8e-1 1.4e-1 1.e-1 1.4e-1 1.4e-1 1.5e-1 1.5e-1 1.4e-1 1.e-1 9.1e- 1.e-1 1.1e-1 9.4e- 7.9e- 5.6e- 7.8e- 6.5e- 1.3e-1 1.3e-1 1.1e-1 9.4e- 70 γ = 1/ CVFEM-SUCV (β= ½) 7.3e- 4.9e-

32 Artificial diffusivity should increase with wave number suggesting poor behavior for FOU /Pe art 1.4 First-Order Upwind (FOU) Second-Order Upwind (FOU) Fromm's Method hird-order Upwind (FOU) QUICK FOU SOU OU QUICK Fromm s Method FEM-SUPG (β opt ) FEM-SUPG (β=½).e. Ο(1) 1. CVFEM-SUPG (β opt ) 1/Pe art FEM - SUPG β = 1/ CVFEM - SUPG β=1/ CVFEM-SUPG (β=½) FEM - SUPG β opt FOU artificial diffusivity acts at ALL wavelengths CVFEM - SUPG β opt x /λ

33 he spectral dependence of the artificial diffusivity does not completely explain its damping effects Given the fundamental solution: x, t = ( x)exp( α k ( ) ) 0 art t Damping of depends on α art and k: x, t t = α k ( x)exp( ( ) α k t) art Similar dependency for Q= t M: ( ) ( Q α k exp α k t) t = art 0 art art Q 0 k = π t = 0 t =.01/α art Q(t)/Q(t=0) k = 4π x tα art

34 Least-Squares Gradient Reconstruction schemes minimize angular dependence of artificial diffusivity art 1/ Pe 0 FOU SOU OU Method Anisotropy 7.8e-.1e-1.1e-1 Diffusivity 6.9e-1 5.1e-1 1.7e-1 QUICK.1e-1.5e LSR(0) art 1/ Pe Fromm s LSR(0) LSR(-1) FEM-SUPG (β opt ) FEM-SUPG (β=½).1e-1 4.5e- 4.8e- 3.8e-1 4.5e-1.6e-1 3.1e-1 6.1e-1 1.7e-1.4e CVFEM-SUCV (β opt ) 3.4e-1 1.6e-1 CVFEM-SUCV (β=½) 4.1e-1.3e-1 70 FEM-SUPG (β = ½)