Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes

Journal of Computational and Applied Mathematics

Volumn 236, Issue 9, 2012, Pages 2317-2337

  Kuzmin, Dmitri ^a  

^a UNIVERSITY OF ERLANGEN NUREMBERG   (Germany)

Author keywords

Anderson acceleration; Discrete maximum principle; Flux correction; Linearity preservation; Slope limiting; Transport equations

Indexed keywords

ANDERSONS; ANISOTROPIC DIFFUSION; ANISOTROPIC DIFFUSION EQUATIONS; CONSISTENT-MASS MATRICES; CONVECTION-DOMINATED; CONVECTIVE FLUX; CONVERGENCE ACCELERATION; CORRECTION SCHEMES; DEFECT CORRECTION; DISCRETE MAXIMUM PRINCIPLES; FINITE ELEMENT; FUNCTION VALUES; GALERKIN DISCRETIZATION; GALERKIN SCHEME; GRADIENT RECOVERY; GRID CONVERGENCE; NEIGHBORING NODES; NON-LINEAR ALGEBRAIC SYSTEM; NONLINEAR CONVERGENCE; QUASI-NEWTON UPDATE; SCALAR TRANSPORT EQUATION; SLOPE LIMITING; TIME-DEPENDENT PROBLEM; TOTAL FLUX; TRANSPORT EQUATION; TRANSPORT PROBLEMS;

ACCELERATION; ALGEBRA; ALGORITHMS; GALERKIN METHODS; MAXIMUM PRINCIPLE; OPTICAL ANISOTROPY; PARTIAL DIFFERENTIAL EQUATIONS; POINT DEFECTS;

CONVERGENCE OF NUMERICAL METHODS;

EID: 84856209194     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2011.11.019     Document Type: Article

References (50)

1. D. Kuzmin, M.J. Shashkov, and D. Svyatskiy A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems J. Comput. Phys. 228 2009 3448 3463
2. K. Lipnikov, M. Shashkov, D. Svyatskiy, and Yu. Vassilevski Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes J. Comput. Phys. 227 2007 492 512
3. V. John, and P. Knobloch On spurious oscillations at layers diminishing (SOLD) methods for convectiondiffusion equations: part Ia review Comput. Methods Appl. Mech. Engrg. 196 1720 2007 2197 2215 (Pubitemid 46238314)
4. Q 1 finite elements Comput. Methods Appl. Mech. Engrg. 197 2008 1997 2014
5. V. John, and E. Schmeyer On finite element methods for 3D time-dependent convectiondiffusion-reaction equations with small diffusion Comput. Methods Appl. Mech. Engrg. 198 2008 475 494
6. J.P. Boris, and D.L. Book Flux-corrected transport: I. SHASTA, a fluid transport algorithm that works J. Comput. Phys. 11 1973 38 69
7. S.T. Zalesak Fully multidimensional flux-corrected transport algorithms for fluids J. Comput. Phys. 31 1979 335 362
8. A. Harten High resolution schemes for hyperbolic conservation laws J. Comput. Phys. 49 1983 357 393
9. P.K. Sweby High resolution schemes using flux limiters for hyperbolic conservation laws SIAM J. Numer. Anal. 21 1984 995 1011 (Pubitemid 14664098)
10. P. Arminjon, A. Dervieux, Construction of TVD-like artificial viscosities on 2-dimensional arbitrary FEM grids, INRIA Research Report, 1111, 1989.
11. R. Lhner, K. Morgan, J. Peraire, and M. Vahdati Finite element flux-corrected transport (FEM-FCT) for the Euler and NavierStokes equations Internat. J. Numer. Methods Fluids 7 1987 1093 1109 (Pubitemid 18522227)
12. V. Selmin, Finite element solution of hyperbolic equations. I. One-dimensional case, INRIA Research Report, 655, 1987.
13. V. Selmin, Finite element solution of hyperbolic equations. II. Two-dimensional case, INRIA Research Report, 708, 1987.
14. R. Lhner Edges, stars, superedges and chains Comput. Methods Appl. Mech. Engrg. 111 1994 255 263
15. P.R.M. Lyra, Unstructured grid adaptive algorithms for fluid dynamics and heat conduction, Ph.D Thesis, University of Wales, Swansea, 1994.
16. J. Peraire, M. Vahdati, J. Peiro, and K. Morgan The construction and behaviour of some unstructured grid algorithms for compressible flows, in: Numerical Methods for Fluid Dynamics, vol. IV 1993 Oxford University Press pp. 221239
17. V. Selmin The node-centred finite volume approach: bridge between finite differences and finite elements Comput. Methods Appl. Mech. Engrg. 102 1993 107 138
18. D. Kuzmin On the design of general-purpose flux limiters for implicit FEM with a consistent mass matrix. I. Scalar convection J. Comput. Phys. 219 2006 513 531 (Pubitemid 44688736)
19. D. Kuzmin Algebraic flux correction for finite element discretizations of coupled systems E. Oate, M. Papadrakakis, B. Schrefler, Computational Methods for Coupled Problems in Science and Engineering II 2007 CIMNE Barcelona 653 656
20. D. Kuzmin, and M. Mller Algebraic flux correction I. scalar conservation laws D. Kuzmin, R. Lhner, S. Turek, Flux-Corrected Transport: Principles, Algorithms, and Applications 2005 Springer 155 206
21. A. Jameson Computational algorithms for aerodynamic analysis and design Appl. Numer. Math. 13 1993 383 422
22. A. Jameson Analysis and design of numerical schemes for gas dynamics 1. artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence Int. J. CFD 4 1995 171 218
23. D. Kuzmin Positive finite element schemes based on the flux-corrected transport procedure K.J. Bathe, Computational Fluid and Solid Mechanics 2001 Elsevier 887 888
24. D. Kuzmin Explicit and implicit FEM-FCT algorithms with flux linearization J. Comput. Phys. 228 2009 2517 2534
25. D. Kuzmin, and S. Turek Flux correction tools for finite elements J. Comput. Phys. 175 2002 525 558
26. J.-C. Carette, H. Deconinck, H. Paillre, and P.L. Roe Multidimensional upwinding: Its relation to finite elements Int. J. Numer. Methods Fluids 20 89 1995 935 955
27. K. Mer Variational analysis of a mixed element/volume scheme with fourth-order viscosity on general triangulations Comput. Methods Appl. Mech. Engrg. 153 1998 45 62
28. D.G. Anderson Iterative procedures for nonlinear integral equations J. Assoc. Comput. Mach. 12 1965 547 560
29. H.W. Walker, and P. Ni Anderson acceleration for fixed-point iterations SIAM J. Numer. Anal. 49 2011 1715 1735
30. V. Eyert A comparative study on methods for convergence acceleration of iterative vector sequences J. Comput. Phys. 124 1996 271 285 (Pubitemid 126170316)
31. H. Fang, and Y. Saad Two classes of multisecant methods for nonlinear acceleration Numer. Linear Algebra Appl. 16 2009 197 221
32. P. Ni, Anderson acceleration of fixed-point iteration with applications to electronic structure computations, Ph.D. Thesis, Worcester Polytechnic Institute, 2009.
33. C.A.J. Fletcher The group finite element formulation Comput. Methods Appl. Mech. Engrg. 37 1983 225 243
34. C.A.J. Fletcher A comparison of finite element and finite difference solutions of the one- and two-dimensional Burgers' equations J. Comput. Phys. 51 1983 159 188
35. D. Kuzmin, A guide to numerical methods for transport equations. University Erlangen-Nuremberg, 2010. Manuscript available online under http://www.mathematik.uni-dortmund.de/~kuzmin/Transport.pdf.
36. T. Barth, and M. Ohlberger Finite Volume Methods: Foundation and Analysis E. Stein, R. de Borst, T.J.R. Hughes, Encyclopedia of Computational Mechanics, Volume 1: Fundamentals 2004 John Wiley & Sons 439 474
37. I. Farag, and R. Horvth Continuous and discrete parabolic operators and their qualitative properties IMA J. Numer. Anal. 29 2009 606 631
38. R.S. Varga Matrix Iterative Analysis 1962 Prentice-Hall, Englewood Cliffs
39. P. Bochev, D. Ridzal, G. Scovazzi, and M. Shashkov Formulation, analysis and numerical study of an optimization-based conservative interpolation (remap) of scalar fields for arbitrary LagrangianfEulerian methods J. Comput. Phys 230 2011 10.1016/j.jcp.2011.03.017
40. R. Lhner Applied CFD Techniques: An Introduction Based on Finite Element Methods 2nd ed. 2008 John Wiley & Sons Chichester
41. M. Arioli, D. Loghin, and A.J. Wathen Stopping criteria for iterations in finite element methods Numer. Math. 99 2006 381 410
42. M. Mller Efficient solution techniques for implicit finite element schemes with flux limiters Int. J. Numer. Methods Fluids 55 2007 611 635
43. H. Luo, J.D. Baum, and R. Lhner A fast, matrix-free implicit method for compressible flows on unstructured grids J. Comput. Phys. 146 1998 664 690 (Pubitemid 128180423)
44. A. Jemcov, and J.P. Maruszewski Algorithm stabilization and acceleration in computational fluid dynamics: exploiting recursive properties of fixed point algorithms R.S. Amano, B. Sundn, Computational Fluid Dynamics and Heat Transfer 2010 WIT Press
45. D.A. Smith, W.F. Ford, and A. Sidi Extrapolation methods for vector sequences SIAM Rev. 29 1987 199 233
46. R.J. LeVeque High-resolution conservative algorithms for advection in incompressible flow SIAM J. Numer. Anal. 33 1996 627 665 (Pubitemid 126431732)
47. J. Donea, and A. Huerta Finite Element Methods for Flow Problems 2003 John Wiley & Sons Chichester
48. M.E. Hubbard Non-oscillatory third order fluctuation splitting schemes for steady scalar conservation laws J. Comput. Phys. 222 2007 740 768
49. M. Gurris, D. Kuzmin, S. Turek, Implicit finite element schemes for the stationary compressible Euler equations, Int. J. Numer. Methods Fluids, in press (doi:10.1002/fld.2532).
50. D. Kuzmin, M. Mller, J.N. Shadid, and M.J. Shashkov Failsafe flux limiting and constrained data projections for equations of gas dynamics J. Comput. Phys. 229 2010 8766 8779