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
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)
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)
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)
1.2 Studied Discretizations – SUPG • Streamline-Upwind Petrov–Galerkin (SUPG) method, [1,2] ◦ stabilization in streamline direction with additional term ∑ ( − ε ∆ u h + b · ∇ u h + cu h − f , y h b · ∇ v h ) K K ∈ T h ◦ a standard parameter choice α , Pe K = | b | h K 2 p | b | ξ ( Pe K ) with ξ ( α ) = coth α − 1 h K y h | K = 2 p ε [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)
1.2 Studied Discretizations – SOLD • SOLD (Spurious Oscillations at Layers Diminishing) method ◦ also called shock capturing methods ◦ adding cross wind stabilization term ( ˜ ε D ∇ u h , ∇ v h ) to SUPG ◦ method from [1,2] I − b ⊗ b if b � = 0 , � diam ( K ) | R h ( u h ) | � | b | 2 D = ε = max ˜ 0 , σ sold − ε 2 | ∇ u h | 0 if b = 0 , ◦ 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)
1.2 Studied Discretizations – CIP • Continuous Interior Penalty (CIP) method from [1] ◦ penalize discontinuities across faces of ∇ u h a ( u h , v h )+ ∑ σ cip h 2 E ( b · [ | ∇ u h | ] E , b · [ | ∇ v h | ] E ) E = ( f , v h ) ∀ v h ∈ V h , E ∈ E h with [ | w | ] E ( x ) : = lim s → 0 ( w ( x + s n E ) − w ( x − s n E )) , 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)
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)
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 � N � N ˜ ∑ Au = f + α i j φ i j 0 ≤ α i j ≤ 1 , j = 1 i = 1 φ i j fluxes ◦ computation of weights α i 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)
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)
1.3 Numerical Studies • Hemker problem with ε = 10 − 4 ◦ reference solution on very fine grid could be computed • P k , Q k , 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)
1.3 Numerical Studies • only P 1 , Q 1 • 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)
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)
1.3 Numerical Studies • error to cut line at y = 1 (tangential to the circle) ◦ least errors: DG, SUPG (for Q 1 ) ◦ 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)
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)
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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