Evolution Galerkin methods for hyperbolic systems in two space dimensions

Mathematics of Computation

Volumn 69, Issue 232, 2000, Pages 1355-1384

  Lukacova Medvid'Ova M ^a,b   Morton, K W ^b   Warnecke, G ^c  

^a OTTO VON GUERICKE UNIVERSITY   (Germany)

^b BRNO UNIVERSITY OF TECHNOLOGY   (Czech Republic)

^c University of Bath   (United Kingdom)

Author keywords

Euler equations; Evolution Galerkin schemes; Genuinely multidimensional schemes; Hyperbolic systems; Wave equation

Indexed keywords

EID: 0034409204     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0025-5718-00-01228-x     Document Type: Article

References (47)

1. J.P. Benqué, G. Labadie, and J. Ronat, A new finite element method for the Navier-Stokes equations coupled with a temperature equation. In T. Kawai, editor, Proceedings of the Fourth International Symposium on Finite Element Methods in Flow Problems, North-Holland, 1982, pp. 295-301.
2. A. Brandt and S. Ta'asan, Multigrid solutions to quasi-elliptic schemes, In Earll. M. Murman and Saul Abarbanel, editors, Progress and Supercomputing in Computational Fluid Dynamics, Proceedings of the U.S.-Israel Workshop 1984, Birkhauser 1985. MR 88m:65185
3. A. Brandt and S. Ta'asan, Multigrid solutions to quasi-elliptic schemes, In Earll. M. Murman and Saul Abarbanel, editors, Progress and Supercomputing in Computational Fluid Dynamics, Proceedings of the U.S.-Israel Workshop 1984, Birkhauser 1985. MR 88m:65185
4. D.S. Butler, The numerical solution of hyperbolic systems of partial differential equations in three independent variables, Proc. Roy. Soc. 255A (1960), 233-252. MR 22:10193
5. D.S. Butler, The numerical solution of hyperbolic systems of partial differential equations in three independent variables, Proc. Roy. Soc. 255A (1960), 233-252. MR 22:10193
6. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. MR 58:25001
7. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. MR 58:25001
8. P.N. Childs and K.W. Morton, Characteristic Galerkin methods for scalar conservation laws in one dimension, SIAM J. Numer. Anal. 27 (1990), 553-594. MR 91e:65115
9. P.N. Childs and K.W. Morton, Characteristic Galerkin methods for scalar conservation laws in one dimension, SIAM J. Numer. Anal. 27 (1990), 553-594. MR 91e:65115
10. M.C. Cline and J.D. Hoffman, Comparison of characteristic schemes for three-dimensional, steady, is entropic flow, AIAA J. 10(11) 1972, 1452-1458.
11. R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience Publishers, 1962. MR 25:4216
12. R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience Publishers, 1962. MR 25:4216
13. J. Douglas and T.F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal. 19 (1982), 871-885. MR 84b:65093
14. J. Douglas and T.F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal. 19 (1982), 871-885. MR 84b:65093
15. M. Fey, Ein echt mehrdimensionales Verfahren zur Lösung der Eulergleichungen, Dissertation, ETH Zürich, 1993.
16. M. Fey and R. Jeltsch, A simple multidimensional Euler scheme, Proceedings of the First European Computational Fluid Dynamics Conference, ECCOMAS'92 vol.I, Ch. Hirsch et.al., editors, Elsevier Science Publishers, Amsterdam, 1992.
17. T.N. Krishnamurti, Numerical integration of primitive equations by a quasi-Lagrangian advective scheme, J. Appl. Meteorology 1 (1962), 508-521.
18. P. Lesaint, Numerical solution of the equation of continuity, In J.J.H. Miller, editor, Topics in Numerical Analysis III, Academic Press, 1977, pp. 199-222. MR 58:31920
19. P. Lesaint, Numerical solution of the equation of continuity, In J.J.H. Miller, editor, Topics in Numerical Analysis III, Academic Press, 1977, pp. 199-222. MR 58:31920
20. R.J. LeVeque, Cartesian grids and rotated difference methods for multi-dimensional flow, Proceedings of the Int. Conf. Sci. Comput. (Hangzhou, China, 1991), Ser. Appl. Math., 1 (1992), pp. 76-85. MR 93g:65119
21. R.J. LeVeque, Cartesian grids and rotated difference methods for multi-dimensional flow, Proceedings of the Int. Conf. Sci. Comput. (Hangzhou, China, 1991), Ser. Appl. Math., 1 (1992), pp. 76-85. MR 93g:65119
22. P. Lin, K.W. Morton, and E. Süli, Euler characteristic Galerkin scheme with recovery, RAIRO Modél. Math. Anal. Numér. 27(7) (1993), 863-894. MR, 94m:65162
23. P. Lin, K.W. Morton, and E. Süli, Euler characteristic Galerkin scheme with recovery, RAIRO Modél. Math. Anal. Numér. 27(7) (1993), 863-894. MR, 94m:65162
24. P. Lin, K.W. Morton, and E. Süli, Characteristic Galerkin schemes for scalar conservation laws in two and three space dimensions, SIAM J. Numer. Anal. 34(2) (1997), 779-796. MR 99b:65118
25. P. Lin, K.W. Morton, and E. Süli, Characteristic Galerkin schemes for scalar conservation laws in two and three space dimensions, SIAM J. Numer. Anal. 34(2) (1997), 779-796. MR 99b:65118
26. M. Lukáčová-Medvid'ová, K.W. Morton, and G. Warnecke, The second order evolution Galerkin schemes for hyperbolic systems, In preparation.
27. M. Lukáčová-Medvid'ová, K.W. Morton, and G. Warnecke, On the evolution Galerkin method for solving multidimensional hyperbolic systems, Proceedings of the Second European Conference on Numerical Mathematics and Advanced Applications (ENUMATH), World Scientific Publishing Company, Singapore, 1998, pp. 445-452.
28. K.W. Morton, Approximation of multidimensional hyperbolic partial differential equations, In I.S. Duff and G.A. Watson, editors, Procedings of the State of the Art in Numerical Analysis Conference, OUP, 1997, pp. 473-502. MR 99e:65142
29. K.W. Morton, Approximation of multidimensional hyperbolic partial differential equations, In I.S. Duff and G.A. Watson, editors, Procedings of the State of the Art in Numerical Analysis Conference, OUP, 1997, pp. 473-502. MR 99e:65142
30. K.W. Morton, Numerical Solution of Convection-Diffusion Problems, Applied Mathematics and Mathematical Computation vol. 12, Chapman & Hall, London, 1996. MR 98b:65004
31. K.W. Morton, Numerical Solution of Convection-Diffusion Problems, Applied Mathematics and Mathematical Computation vol. 12, Chapman & Hall, London, 1996. MR 98b:65004
32. K.W. Morton, On the analysis of finite volume methods for evolutionary problems, SIAM J. Numer. Anal. 35(6) (1998), 2195-2222. MR 99m:65172
33. K.W. Morton, On the analysis of finite volume methods for evolutionary problems, SIAM J. Numer. Anal. 35(6) (1998), 2195-2222. MR 99m:65172
34. K.W. Morton and P.L. Roe, Vorticity-preserving Lax-Wendroff type schemes for the system wave equation, submitted for publication.
35. S. Ostkamp, Multidimensional characterisitic Galerkin schemes and evolution operators for hyperbolic systems, PhD thesis, Universität Hannover, 1995. MR 97e:65098
36. S. Ostkamp, Multidimensional characterisitic Galerkin schemes and evolution operators for hyperbolic systems, PhD thesis, Universität Hannover, 1995. MR 97e:65098
37. S. Ostkamp, Multidimensional characterisitic Galerkin schemes and evolution operators for hyperbolic systems, Math. Meth. Appl. Sci. 20 (1997), 1111-1125. MR 98g:65093
38. S. Ostkamp, Multidimensional characterisitic Galerkin schemes and evolution operators for hyperbolic systems, Math. Meth. Appl. Sci. 20 (1997), 1111-1125. MR 98g:65093
39. O. Pironneau, On the transport-diffusion algorithm and its application to the Navier-Stokes equations, Numer. Math. 38 (1982), 309-332. MR 83d:65258
40. O. Pironneau, On the transport-diffusion algorithm and its application to the Navier-Stokes equations, Numer. Math. 38 (1982), 309-332. MR 83d:65258
41. P. Prasad and R. Ravindran, Canonical form of a quasilinear hyperbolic system of first order equations, J. Math. Phys. Sci. 18(4) (1984), 361-364. MR 87c:35108
42. P. Prasad and R. Ravindran, Canonical form of a quasilinear hyperbolic system of first order equations, J. Math. Phys. Sci. 18(4) (1984), 361-364. MR 87c:35108
43. A.S. Reddy, V.G. Tikekar, and P. Prasad, Numerical solution of hyperbolic equations by method of bicharactenstics, Journal of Mathematical and Physical Sciences 16(6) (1982), 575-603. MR 84h:65094
44. A.S. Reddy, V.G. Tikekar, and P. Prasad, Numerical solution of hyperbolic equations by method of bicharactenstics, Journal of Mathematical and Physical Sciences 16(6) (1982), 575-603. MR 84h:65094
45. A. Staniforth and J. Côté, Semi-Lagrangian integration schemes and their application to environmental flows, Monthly Weather Rev. 119(9) (1991), 2206-2223.
46. E. Süli, Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations, Numer. Math. 53 (1988), 459-483.
47. M. Tanguay, A. Simard, and A. Staniforth, A three-dimensional semi-Lagrangian scheme for the Canadian regional finite-element forecast model, Monthly Weather Rev. 117 (1989), 1861-1871.