A finite difference method for symmetric positive differential equations

Liu, Jinn Liang

doi:10.1090/S0025-5718-1994-1208839-5

A finite difference method for symmetric positive differential equations

Author: Jinn Liang Liu
Journal: Math. Comp. 62 (1994), 105-118
MSC: Primary 65N06
DOI: https://doi.org/10.1090/S0025-5718-1994-1208839-5
MathSciNet review: 1208839
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A finite difference method is developed for solving symmetric positive differential equations in the sense of Friedrichs. The method is applicable to partial differential equations of mixed type with more general boundary conditions. The method is shown to have a convergence rate of $O({h^{1/2}})$ , h being the size of mesh grid. Some numerical results are presented for a model problem of forward-backward heat equations.

[1] A. K. Aziz and J.-L. Liu, A weighted least squares method for the backward-forward heat equation, SIAM J. Numer. Anal. 28 (1991), no. 1, 156–167. MR 1083329, https://doi.org/10.1137/0728008
[2] M. S. Baouendi and P. Grisvard, Sur une équation d’évolution changeant de type, J. Functional Analysis 2 (1968), 352–367 (French). MR 0252817
[3] Richard Beals, On an equation of mixed type from electron scattering theory, J. Math. Anal. Appl. 58 (1977), no. 1, 32–45. MR 0492921, https://doi.org/10.1016/0022-247X(77)90225-6
[4] C. K. Chu, Type-insensitive finite difference schemes, Ph.D. Thesis, New York University, 1958.
[5] K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 333–418. MR 0100718, https://doi.org/10.1002/cpa.3160110306
[6] Jerome A. Goldstein and Tapas Mazumdar, A heat equation in which the diffusion coefficient changes sign, J. Math. Anal. Appl. 103 (1984), no. 2, 533–564. MR 762573, https://doi.org/10.1016/0022-247X(84)90145-8
[7] Theodore Katsanis, Numerical solution of symmetric positive differential equations, Math. Comp. 22 (1968), 763–783. MR 0245214, https://doi.org/10.1090/S0025-5718-1968-0245214-9
[8] T. LaRosa, The propagation of an electron beam through the solar corona, Ph.D. Thesis, Dept. of Physics and Astronomy, University of Maryland, 1986.
[9] P. Lesaint, Finite element methods for symmetric hyperbolic equations, Numer. Math. 21 (1973/74), 244–255. MR 0341902, https://doi.org/10.1007/BF01436628
[10] P. Lesaint and P.-A. Raviart, Finite element collocation methods for first-order systems, Math. Comp. 33 (1979), no. 147, 891–918. MR 528046, https://doi.org/10.1090/S0025-5718-1979-0528046-0
[11] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). MR 0259693
[12] V. Vanaja and R. B. Kellogg, Iterative methods for a forward-backward heat equation, SIAM J. Numer. Anal. 27 (1990), no. 3, 622–635. MR 1041255, https://doi.org/10.1137/0727038