Energy-conserving norms for the solution of hyperbolic systems of partial differential equations
Authors:
Max D. Gunzburger and Robert J. Plemmons
Journal:
Math. Comp. 33 (1979), 1-10
MSC:
Primary 35L50; Secondary 65N20
DOI:
https://doi.org/10.1090/S0025-5718-1979-0514807-0
MathSciNet review:
514807
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: The problem of finding an energy conserving norm for the solution of the hyperbolic system of partial differential equations $\partial u/\partial t = A\partial u/\partial x$, subject to boundary conditions, is reduced to the problem of characterizing those matrices appearing in the boundary conditions which satisfy two specific matrix equations. Necessary and sufficient conditions on the coefficient matrix A and the matrices appearing in boundary conditions are derived for an energy conserving norm to exist. Thus, these conditions serve as tests on a given system which determine whether or not the solution will have its energy conserved in some norm. In addition, some examples of specific systems and boundary conditions are provided.
- Charles G. Cullen, Matrices and linear transformations, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0197472
- C. R. DePrima and C. R. Johnson, The range of $A^{-1}A^{\ast } $ in ${\rm GL}(n,\,C)$, Linear Algebra Appl. 9 (1974), 209β222. MR 361862, DOI https://doi.org/10.1016/0024-3795%2874%2990039-1
- Ky Fan, On strictly dissipative matrices, Linear Algebra Appl. 9 (1974), 223β241. MR 369393, DOI https://doi.org/10.1016/0024-3795%2874%2990040-8
- Max D. Gunzburger, On the stability of Galerkin methods for initial-boundary value problems for hyperbolic systems, Math. Comp. 31 (1977), no. 139, 661β675. MR 436624, DOI https://doi.org/10.1090/S0025-5718-1977-0436624-0 H. O. KREISS AND J. OLIGER, Methods for the Approximate Solution Time Dependent Problems, Global Atmospheric Research Programme Publ. Ser., no. 10, Geneva, 1973.
- Olga Taussky, Positive-definite matrices and their role in the study of the characteristic roots of general matrices, Advances in Math. 2 (1968), 175β186. MR 227200, DOI https://doi.org/10.1016/0001-8708%2868%2990020-0
Retrieve articles in Mathematics of Computation with MSC: 35L50, 65N20
Retrieve articles in all journals with MSC: 35L50, 65N20
Additional Information
Article copyright:
© Copyright 1979
American Mathematical Society