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 , 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), 156-167. MR 1083329 (92a:65333)
- [2] M. S. Baouendi and P. Grisvard, Sur une équation d'évolution changeant de type, J. Funct. Anal. 2 (1968), 352-367. MR 0252817 (40:6034)
- [3] R. Beals, On an equation of mixed type from electron scattering theory, J. Math. Anal. Appl. 58 (1977), 32-45. MR 0492921 (58:11970)
- [4] C. K. Chu, Type-insensitive finite difference schemes, Ph.D. Thesis, New York University, 1958.
- [5] K. O. Friedrichs, Symmetric positive differential equations, Comm. Pure Appl. Math. 11 (1958), 333-418. MR 0100718 (20:7147)
- [6] J. A. Goldstein and T. Mazumdar, A heat equation in which the diffusion coefficient changes sign, J. Math. Anal. Appl. 103 (1984), 533-564. MR 762573 (86g:35091)
- [7] T. Katsanis, Numerical solution of symmetric positive differential equations, Math. Comp. 22 (1968), 763-783. MR 0245214 (39:6526)
- [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), 244-255. MR 0341902 (49:6648)
- [10] P. Lesaint and P. A. Raviart, Finite element collocation methods for first order systems, Math. Comp. 33 (1979), 891-918. MR 528046 (80d:65118)
- [11] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limits non linéaires, Gauthier-Villars, Paris, 1969, pp. 337-343. MR 0259693 (41:4326)
- [12] V. Vanaja and R. B. Kellogg, Iterative methods for a forward-backward heat equation, SIAM J. Numer. Anal. 27 (1990), 622-635. MR 1041255 (91d:65158)
Retrieve articles in Mathematics of Computation with MSC: 65N06
Retrieve articles in all journals with MSC: 65N06
Additional Information
DOI:
https://doi.org/10.1090/S0025-5718-1994-1208839-5
Keywords:
Finite difference method,
Friedrichs's positive systems,
error estimates
Article copyright:
© Copyright 1994
American Mathematical Society