On Recent Experience With Discretizations of Convection-Diffusion - - PowerPoint PPT Presentation

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On Recent Experience With Discretizations of Convection-Diffusion - - PowerPoint PPT Presentation

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Weierstrass Institute for Applied Analysis and Stochastics On Recent Experience With Discretizations of Convection-Diffusion Equations Volker John (WIAS Berlin and Free University of Berlin) Mohrenstrasse 39 10117 Berlin Germany Tel.

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Weierstrass Institute for Applied Analysis and Stochastics

On Recent Experience With Discretizations of Convection-Diffusion Equations

Volker John (WIAS Berlin and Free University of Berlin)

Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de · Workshop Numerical Analysis for Singularly Perturbed Problems, dedicated to the 60-th birthday of Martin Stynes, November 17, 2011

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Outline of the talk

1 Steady-State Convection-Diffusion Equations Motivation Studied Discretizations Numerical Studies Summary 2 Time-Dependent Convection-Diffusion Equations Motivation Studied Discretizations Numerical Studies Summary

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 2 (27)

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  • 1. Steady-State Convection-Diffusion Equations∗

Talk in Magdeburg 2000

∗ joint work with M. Augustin (Kaiserslautern), A. Caiazzo (WIAS), A. Fiebach (WIAS), J. Fuhrmann (WIAS), A. Linke

(WIAS), R. Umla (Imperial College London)

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 3 (27)

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1.1 Motivation

  • scalar convection-diffusion equation

−ε∆u+b·∇u = f in Ω + boundary conditions

  • appear in many applications (energy and mass balances)
  • in convection-dominated regime stabilized discretization necessary
  • many stabilized methods proposed in literature
  • goals
  • consider methods based on finite element and finite volume ideas
  • evalutate quantities which are of interest in applications

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 4 (27)

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1.2 Studied Discretizations – SUPG

  • Streamline-Upwind Petrov–Galerkin (SUPG) method, [1,2]
  • stabilization in streamline direction with additional term

∑

K∈Th

(−ε ∆uh +b·∇uh +cuh − f,yh b·∇vh)K

  • a standard parameter choice

yh|K = hK 2 p|b| ξ(PeK) with ξ(α) = cothα − 1 α , PeK = |b|hK 2 pε

[1] Hughes, Brooks: Finite Element Methods for Convection Dominated Flows, 19 – 35, 1979 [2] Brooks, Hughes: Comput. Methods Appl. Mech. Engrg. 32, 199 – 259, 1982

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 5 (27)

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1.2 Studied Discretizations – SOLD

  • SOLD (Spurious Oscillations at Layers Diminishing) method
  • also called shock capturing methods
  • adding cross wind stabilization term (˜

εD∇uh,∇vh) to SUPG

  • method from [1,2]

D =    I − b⊗b |b|2 if b = 0, if b = 0, ˜ ε = max

  • 0,σsold

diam(K)|Rh(uh)| 2|∇uh| −ε

  • best method in extensive numerical studies in [3,4]
  • user-chosen stabilization parameter
  • nonlinear

[1] Codina: Comput. Methods Appl. Mech. Engrg. 110, 325 – 342, 1993 [2] Knopp, Lube, Rapin: Comput. Methods Appl. Mech. Engrg. 191, 2997 – 3013, 2002 [3] J., Knobloch: Comput. Methods Appl. Mech. Engrg. 196, 2197 – 2215, 2007 [4] J., Knobloch: Comput. Methods Appl. Mech. Engrg. 197, 1997 – 2014, 2008

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 6 (27)

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1.2 Studied Discretizations – CIP

  • Continuous Interior Penalty (CIP) method from [1]
  • penalize discontinuities across faces of ∇uh

a(uh,vh)+ ∑

E∈Eh

σciph2

E (b·[|∇uh|]E,b·[|∇vh|]E)E = ( f,vh)

∀vh ∈ Vh, with [|w|]E(x) := lim

s→0(w(x+snE)−w(x−snE)),

x ∈ E

  • user-chosen stabilization parameter

[1] Burman, Hansbo: Comput. Methods Appl. Mech. Engrg. 193, 1437 – 1453, 2004

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 7 (27)

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1.2 Studied Discretizations – DG

  • Discontinuous Galerkin (DG) method from [1]
  • discontinuous finite element spaces
  • stabilization by weakly imposed continuity
  • user-chosen stabilization parameter

[1] Kanschat: Discontinuous Galerkin Methods for Viscous Incompressible Flow, Teubner, 2007

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 8 (27)

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1.2 Studied Discretizations – FEMTVD

  • Total Variation Diminishing Finite Element Method (FEMTVD) from [1]
  • manipulations of the algebraic scheme

Au = f = ⇒ ˜ Au = f A – from SUPG (or other) discretization, ˜ A – M–matrix

  • new scheme too diffusive
  • remove diffusion by manipulating right hand side

˜ Au = f +

  • N

∑

j=1

αi jφi j N

i=1

, 0 ≤ αi j ≤ 1 φi j fluxes

  • computation of weights αi j is nonlinear process

[1] Kuzmin: Proceedings of the ECCOMAS Conference Computational Methods for Coupled Problems in Science and Engineering, 2007

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 9 (27)

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1.2 Studied Discretizations – Exponetial Fitted FVM

  • Exponentially Fitted Voronoi Box Finite Volume Method from [1]
  • on Delaunay meshes
  • leads to convection-dominated 1D equations on lines connecting

the control volumes

  • solved with Il’in–Allen–Southwell scheme (sometimes called

Scharfetter–Gummel scheme)

  • local maximum principle holds

[1] Fuhrmann, Langmach: Appl. Numer. Math., 201 – 230, 2001

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 10 (27)

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1.3 Numerical Studies

  • Hemker problem with ε = 10−4
  • reference solution on very fine grid could be computed
  • P

k, Qk, k = 1,2,3

  • critera
  • under- and overshoots
  • smearing of interior layers
  • errors to cut lines from the reference solution
  • computing times vs. quality measures

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 11 (27)

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1.3 Numerical Studies

  • only P

1, Q1

  • under- and overshoots
  • no under- and overshoots: FVM, FEMTVD
  • very large under- and overshoots: SUPG, CIP

, DG (for P

1 not in

diagram)

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 12 (27)

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1.3 Numerical Studies

  • smearing of interior layer
  • least smearing: SUPG, DG
  • very large smearing: FVM, SOLD, CIP

, FEMTVD

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 13 (27)

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1.3 Numerical Studies

  • error to cut line at y = 1 (tangential to the circle)
  • least errors: DG, SUPG (for Q1)
  • large errors: CIP

, FEMTVD, FVM

  • FEMTVD has spurious oscillations

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 14 (27)

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1.3 Numerical Studies

  • computing time vs. quality measures (smearing, cut lines)

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 15 (27)

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1.4 Summary

  • none of the methods optimal
  • nonlinear schemes (SOLD, FEMTVD) too time-consuming
  • advices
  • no under- and overshoots important: use FVM, good (aligned) grid

necessary

  • sharp layers important, under- and overshoots can be tolerated:

SUPG often good choice

  • modern schemes seldom beneficial compared with classical schemes

(FVM, SUPG)

  • urgent need to construct better methods for discretizing

convection-dominated equations

  • details in [1]

[1] Augustin, Caiazzo, Fiebach, Fuhrmann, J., Linke, Umla, Comput. Methods Appl. Mech. Engrg. 200, 3395 – 3409, 2011

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 16 (27)

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  • 2. Time-Dependent Convection-Diffusion Equations‡

Visit in Colbitz 2000

‡ joint work with J. Novo (Madrid) On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 17 (27)

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2.1 Motivation

  • scalar convection-diffusion equation

ut −ε∆u+b·∇u+cu = f in (0,T]×Ω + initial & boundary conditions

  • appear in many applications (energy and mass balances)
  • in convection-dominated regime stabilized discretization necessary
  • our applications: often simple domains in 4D or 5D
  • goals
  • consider methods based on finite element and finite difference

ideas

  • evalutate quantities which are of interest in applications

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 18 (27)

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2.2 Studied Discretizations – FDM

  • TVD Runge–Kutta methods
  • method of Heun (2nd order)
  • optimal 3rd order method
  • discretizations of convective term
  • simple upwinding
  • 3rd order ENO scheme
  • 5th order WENO scheme [1]

[1] Jiang, Shu: J. Comput. Phys. 126, 202 – 228, 1996

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 19 (27)

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2.2 Studied Discretizations – FEM

  • Crank–Nicolson scheme
  • Finite Element Method Flux Corrected Transport (FEM-FCT)
  • works on the algebraic level, similar to FEMTVD
  • linear [1] and nonlinear [2] version used
  • best method in competetive studies [3]
  • standard version (matrix assembling each time step)
  • group fem version (matrix built by matrix-vector multiplications) [4]

[1] Kuzmin: J. Comput. Phys. 228, 2517 – 2534, 2009 [2] Kuzmin, Möller: in Flux-Corrected Transport: Principles, Algorithms and Applications, 155 – 206, 2005 [3] J., Schmeyer: Comput. Methods Appl. Mech. Engrg. 198, 475 – 494, 2008 [4] Fletcher: Int. J. Numer. Methods Fluids 37, 225 – 243, 1983

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 20 (27)

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2.3 Numerical Studies

  • rotating body problem

left: linear FEM-FCT right: nonlinear FEM- FCT left: 3rd RK + ENO right: 3rd RK + WENO

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 21 (27)

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2.3 Numerical Studies

  • under- and overshoots
  • none for FEM-FCT
  • small for ENO and WENO
  • computing times (in sec.)
  • GMRES + Jacobi preconditioner
  • Anderson acceleration for fixed point interation

ENO WENO

  • lin. FEM-FCT
  • lin. GFEM-FCT

nlin FEM-FCT nlin GFEM-FCT 48 86 237 111 1025 890

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 22 (27)

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2.3 Numerical Studies

  • transport of a species through a 3D domain
  • cut planes with inlet (left) and outlet (right)

nonlinear FEM-FCT 3rd RK + WENO

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 23 (27)

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2.3 Numerical Studies

  • solutions at the center of the outlet
  • computing times (in sec.)

ENO WENO

  • lin. FEM-FCT
  • lin. GFEM-FCT

nlin FEM-FCT nlin GFEM-FCT 70 119 926 264 2265 1606

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 24 (27)

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2.3 Numerical Studies

  • aligned grids: preferable in many problems
  • transport of an impulse (academic example)
  • ENO
  • nonlinear GFEM-FCT
  • transport of a log-normal inlet (appears in population balance systems)
  • ENO
  • linear GFEM-FCT

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 25 (27)

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2.3 Numerical Studies

  • aligned grids: preferable in many problems
  • transport of an impulse (academic example)
  • ENO
  • nonlinear GFEM-FCT
  • transport of a log-normal inlet (appears in population balance systems)
  • ENO
  • linear GFEM-FCT
  • FEM-FCT very diffusive
  • reason: method is multi-dimensional, but problem is essentially
  • ne-dimensional
  • cure: use one-dimensional version of FEM-FCT schemes
  • difficulty: distinguish in practice where one-dimensional and where

multi-dimensional version should be applied

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 25 (27)

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2.4 Summary

  • if under- or overshoots cannot be tolerated
  • computing time is strong issue: simple upwinding FDM
  • computing time and accuracy important: linear group FEM-FCT
  • accuracy is strong issue: nonlinear group FEM-FCT
  • small under- and overshoots can be tolerated
  • computing time is strong issue: 3rd order RK + ENO
  • otherwise: 3rd order RK + WENO
  • FEM-FCT schemes
  • use always group version
  • very diffusive on aligned grids
  • details in [1]

[1] J., Novo, J. Comput. Phys. , accepted, 2011

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 26 (27)

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Happy Birthday, Martin !

Wandertag 1998

On Recent Experience With Discretizations of Convection-Diffusion Equations · Workshop Numerical Analysis for Singularly Perturbed Problems, dedi- cated to the 60-th birthday of Martin Stynes, November 17, 2011 · Page 27 (27)