Energy-conserving norms for the solution of hyperbolic systems of partial differential equations
Abstract: The problem of finding an energy conserving norm for the solution of the hyperbolic system of partial differential equations , 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⁻¹𝐴* in 𝐺𝐿(𝑛,𝐶), Linear Algebra and Appl. 9 (1974), 209–222. MR 0361862
-  Ky Fan, On strictly dissipative matrices, Linear Algebra and Appl. 9 (1974), 223–241. MR 0369393
-  Max D. Gunzburger, On the stability of Galerkin methods for initial-boundary value problems for hyperbolic systems, Math. Comp. 31 (1977), no. 139, 661–675. MR 0436624, 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), 175–186. MR 0227200
- C. G. CULLEN, Matrices and Linear Transformation, Addison-Wesley, Reading, Mass., 1966. MR 0197472 (33:5637)
- C. DePRIMA AND C. JOHNSON, "The range of in ," Linear Algebra and Appl., v. 9, 1974, pp. 209-222. MR 0361862 (50:14305)
- KY FAN, "On strictly dissipative matrices," Linear Algebra and Appl., v. 9, 1974, pp. 223-241. MR 0369393 (51:5626)
- M. D. GUNZBURGER, "On the stability of Galerkin methods for initial boundary value problems for hyperbolic systems," Math. Comp., v. 31, 1977, pp. 661-675. MR 0436624 (55:9567)
- H. O. KREISS AND J. OLIGER, Methods for the Approximate Solution Time Dependent Problems, Global Atmospheric Research Programme Publ. Ser., no. 10, Geneva, 1973.
- O. TAUSSKY, "Positive definite matrices and their role in the study of characteristic roots of general matrices," Advances in Math., v. 2, 1968, pp. 175-186. MR 0227200 (37:2785)