Stability of a cartesian grid projection method for zero froude number shallow water flows

Numerische Mathematik

Volumn 113, Issue 1, 2009, Pages 123-161

  Vater, Stefan ^a   Klein, Rupert ^a  

^a FREIE UNIVERSITÄT BERLIN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 67650668216     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-009-0224-8     Document Type: Article

References (41)

1. A.S. Almgren J.B. Bell P. Colella L.H. Howell M.L. Welcome 1998 A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations J. Comput. Phys. 142 1 1 46 (Pubitemid 128344797)
2. A.S. Almgren J.B. Bell W.Y. Crutchfield 2000 Approximate projection methods: Part I. inviscid analysis SIAM J. Sci. Comput. 22 4 1139 1159
3. A.S. Almgren J.B. Bell W.G. Szymczak 1996 A numerical method for the incompressible Navier-Stokes equations based on an approximate projection SIAM J. Sci. Comput. 17 2 358 369 (Pubitemid 126610882)
4. L. Angermann 2003 Node-centered finite volume schemes and primal-dual mixed formulations Commun. Appl. Anal. 7 4 529 566 (Pubitemid 37457684)
5. I. Babuška 1971 Error-bounds for finite element method Numerische Mathematik 16 322 333
6. J.B. Bell P. Colella H.M. Glaz 1989 A second-order projection method for the incompressible Navier-Stokes equations J. Comput. Phys. 85 2 257 283
7. J.B. Bell D.L. Marcus 1992 A second-order projection method for variable-density flows J. Comput. Phys. 101 334 348
8. C. Bernardi C. Canuto Y. Maday 1988 Generalized inf-sup conditions for the Chebyshev spectral approximation of the Stokes problem SIAM J Numer Anal 25 6 1237 1271
9. Braess D.: Finite Elemente: Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie, 3rd edn. Springer, Berlin (2003)
10. F. Brezzi 1974 On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers RAIRO Analyse Numérique 8 129 151
11. A.J. Chorin 1968 Numerical solution of the Navier-Stokes equations Math. Comput. 22 104 745 762
12. R. Courant K.O. Friedrichs H. Lewy 1928 Über die partiellen Differenzengleichungen der mathematischen Physik Mathematische Annalen 100 32 74
13. Geratz, K.J.: Erweiterung eines Godunov-Typ-Verfahrens für zwei-dimensionale kompressible Strömungen auf die Fälle kleiner und verschwindender Machzahl. PhD dissertation, Rheinisch-Westfälische Technische Hochschule Aachen (1997)
14. Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, 2 edn. Springer, Heidelberg (1983)
15. Girault V., Raviart P.A.: Finite Element Methods for Navier-Stokes Equations, Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)
16. P.M. Gresho 1990 On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 1: Theory Int J Numer Methods Fluids 11 5 587 620
17. P.M. Gresho S.T. Chan 1990 On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 2: Implementation Int. J. Numer. Methods Fluids 11 5 621 659
18. J.L. Guermond P. Minev J. Shen 2006 An overview of projection methods for incompressible flows Comput. Methods Appl. Mech. Eng. 195 44-47 6011 6045 (Pubitemid 44081046)
19. F.H. Harlow J.E. Welch 1965 Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface Phys. Fluids 8 12 2182 2189
20. S. Klainerman A. Majda 1981 Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids Commun. Pure Appl. Math. 34 481 524
21. R. Klein 1995 Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I: one-dimensional flow J. Comput. Phys. 121 213 237
22. LeVeque R.J.: Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, vol. 31. Cambridge University Press, Cambridge (2002)
23. M.L. Minion 1996 A projection method for locally refined grids J. Comput. Phys. 127 1 158 178 (Pubitemid 126162335)
24. C.D. Munz S. Roller R. Klein K.J. Geratz 2003 The extension of incompressible flow solvers to the weakly compressible regime Comput. Fluids 32 2 173 196
25. R.A. Nicolaïdes 1982 Existence, uniqueness and approximation for generalized saddle point problems SIAM J. Numer. Anal. 19 2 349 357
26. M. Oevermann R. Klein 2006 A cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces J. Comput. Phys. 219 2 749 769 (Pubitemid 44688747)
27. S. Osher 1985 Convergence of generalized MUSCL schemes SIAM J. Numer. Anal. 22 5 947 961
28. T. Schneider N. Botta K.J. Geratz R. Klein 1999 Extension of finite volume compressible flow solvers to multi-dimensional, variable density zero Mach number flows J. Comput. Phys. 155 248 286
29. S. Schochet 1994 Fast singular limits of hyperbolic PDEs J. Differ. Equ. 114 2 476 512
30. S. Schochet 2005 The mathematical theory of low Mach number flows RAIRO Modélisation Mathématique et Analyse Numérique 39 3 441 458 (Pubitemid 40977974)
31. Schulz-Rinne, C.W.: The Riemann problem for two-dimensional gas dynamics and new limiters for high-order schemes. PhD dissertation, Eidgenössische Technische Fachhochschule (ETH) Zürich. Diss. ETH No. 10297 (1993)
32. C.W. Shu S. Osher 1988 Efficient implementation of essentially non-oscillatory shock-capturing schemes J. Comput. Phys. 77 2 439 471
33. G. Strang 1968 On the construction and comparison of difference schemes SIAM J. Numer. Anal. 5 3 506 517
34. E. Süli 1991 Convergence of finite volume schemes for Poisson's equation on nonuniform meshes SIAM J. Numer. Anal. 28 5 1419 1430 (Pubitemid 21710235)
35. R. Temam 1968 Une méthode d'approximation de la solution des équations de Navier-Stokes Bulletin de la Société Mathématique de France 96 115 152
36. J.M. Thomas D. Trujillo 1999 Mixed finite volume methods Int. J. Numer. Methods Eng. 46 9 1351 1366 (Pubitemid 30532708)
37. G.D. van Albada B. van Leer W.W. Roberts Jr. 1982 A comparative study of computational methods in cosmic gas dynamics Astron. Astrophys. 108 1 76 84
38. H.A. van der Vorst 1992 Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems SIAM J. Sci. Stat. Comput. 13 2 631 644
39. J. van Kan 1986 A second-order accurate pressure-correction scheme for viscous incompressible flow SIAM J. Sci. Stat. Comput. 7 3 870 891
40. B. van Leer 1979 Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method J. Comput. Phys. 32 1 101 136
41. Vater, S.: A new projection method for the zero Froude number shallow water equations. PIK Report 97, Potsdam Institute for Climate Impact Research (2005)