Evolution Galerkin methods for hyperbolic systems in two space dimensions

Lukáčová-Medvid’ová, M.; Morton, K.; Warnecke, G.

doi:10.1090/S0025-5718-00-01228-X

Evolution Galerkin methods for hyperbolic systems in two space dimensions
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by M. Lukáčová-Medvid’ová, K. W. Morton and G. Warnecke PDF

Math. Comp. 69 (2000), 1355-1384 Request permission

Abstract:

The subject of the paper is the analysis of three new evolution Galerkin schemes for a system of hyperbolic equations, and particularly for the wave equation system. The aim is to construct methods which take into account all of the infinitely many directions of propagation of bicharacteristics. The main idea of the evolution Galerkin methods is the following: the initial function is evolved using the characteristic cone and then projected onto a finite element space. A numerical comparison is given of the new methods with already existing methods, both those based on the use of bicharacteristics as well as commonly used finite difference and finite volume methods. We discuss the stability properties of the schemes and derive error estimates.

References

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.
Achi Brandt and Shlomo Ta’asan, Multigrid solutions to quasi-elliptic schemes, Progress and supercomputing in computational fluid dynamics (Jerusalem, 1984) Progr. Sci. Comput., vol. 6, Birkhäuser Boston, Boston, MA, 1985, pp. 235–255. MR 935156
D. S. Butler, The numerical solution of hyperbolic systems of partial differential equations in three independent variables, Proc. Roy. Soc. London Ser. A 255 (1960), 232–252. MR 119432, DOI 10.1098/rspa.1960.0065
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
P. N. Childs and K. W. Morton, Characteristic Galerkin methods for scalar conservation laws in one dimension, SIAM J. Numer. Anal. 27 (1990), no. 3, 553–594. MR 1041252, DOI 10.1137/0727035
M.C. Cline and J.D. Hoffman, Comparison of characteristic schemes for three-dimensional, steady, isentropic flow, AIAA J. 10(11) 1972, 1452–1458.
R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II: Partial differential equations, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. (Vol. II by R. Courant.). MR 0140802
Jim Douglas Jr. and Thomas 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), no. 5, 871–885. MR 672564, DOI 10.1137/0719063
M. Fey, Ein echt mehrdimensionales Verfahren zur Lösung der Eulergleichungen, Dissertation, ETH Zürich, 1993.
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.
T.N. Krishnamurti, Numerical integration of primitive equations by a quasi-Lagrangian advective scheme, J. Appl. Meteorology 1 (1962), 508–521.
P. Lesaint, Numerical solution of the equation of continuity, Topics in numerical analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976) Academic Press, London, 1977, pp. 199–222. MR 0658144
Randall J. LeVeque, Cartesian grids and rotated difference methods for multi-dimensional flow, Proceedings of International Conference on Scientific Computation (Hangzhou, 1991) Ser. Appl. Math., vol. 1, World Sci. Publ., River Edge, NJ, 1992, pp. 76–85. MR 1184846
Peixiong Lin, K. W. Morton, and E. Süli, Euler characteristic Galerkin scheme with recovery, RAIRO Modél. Math. Anal. Numér. 27 (1993), no. 7, 863–894 (English, with English and French summaries). MR 1249456, DOI 10.1051/m2an/1993270708631
Peixiong 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 (1997), no. 2, 779–796. MR 1442938, DOI 10.1137/S0036142993259299
M. Lukáčová-Medvid’ová, K.W. Morton, and G. Warnecke, The second order evolution Galerkin schemes for hyperbolic systems, In preparation.
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.
K. W. Morton, Approximation of multidimensional hyperbolic partial differential equations, The state of the art in numerical analysis (York, 1996) Inst. Math. Appl. Conf. Ser. New Ser., vol. 63, Oxford Univ. Press, New York, 1997, pp. 473–502. MR 1628357
K. W. Morton, Numerical solution of convection-diffusion problems, Applied Mathematics and Mathematical Computation, vol. 12, Chapman & Hall, London, 1996. MR 1445295
K. W. Morton, On the analysis of finite volume methods for evolutionary problems, SIAM J. Numer. Anal. 35 (1998), no. 6, 2195–2222. MR 1655843, DOI 10.1137/S0036142997316967
K.W. Morton and P.L. Roe, Vorticity-preserving Lax-Wendroff type schemes for the system wave equation, submitted for publication.
Stella Ostkamp, Multidimensional characteristic Galerkin schemes and evolution operators for hyperbolic systems, DLR-Forschungsbericht [DLR-Research Report], vol. 95, Deutsche Forschungsanstalt für Luft- und Raumfahrt, Cologne, 1995 (English, with English and German summaries). Dissertation, Universität Hannover, Hannover, 1995. MR 1361170
S. Ostkamp, Multidimensional characteristic Galerkin methods for hyperbolic systems, Math. Methods Appl. Sci. 20 (1997), no. 13, 1111–1125. MR 1465396, DOI 10.1002/(SICI)1099-1476(19970910)20:13<1111::AID-MMA903>3.0.CO;2-1
O. Pironneau, On the transport-diffusion algorithm and its applications to the Navier-Stokes equations, Numer. Math. 38 (1981/82), no. 3, 309–332. MR 654100, DOI 10.1007/BF01396435
Phoolan Prasad and Renuka Ravindran, Canonical form of a quasilinear hyperbolic system of first order equations, J. Math. Phys. Sci. 18 (1984), no. 4, 361–364. MR 803963
A. Sivasankara Reddy, V. G. Tikekar, and Phoolan Prasad, Numerical solution of hyperbolic equations by method of bicharacteristics, J. Math. Phys. Sci. 16 (1982), no. 6, 575–603. MR 700791
A. Staniforth and J. Côté, Semi-Lagrangian integration schemes and their application to environmental flows, Monthly Weather Rev. 119(9) (1991), 2206–2223.
E. Süli, Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations, Numer. Math. 53 (1988), 459–483.
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.