Abstract
We approximate the Cauchy problem by a problem in a bounded domain Ω R =(−R,R) withR>0 sufficiently large, and the boundary conditions on ∂Ω R are imposed in terms of the far field behavior of solutions to the Cauchy problem. Then we solve this approximate problem by the finite element method for the spatial variable and the difference method for the time variable. Moreover a coupled numerical scheme for the Cauchy problem is presented. The error estimates are established.
Zusammenfassung
Wir approximieren das Cauchy-Problem durch ein Problem in einem beschränkten Gebiet Ω R =(−R,R) mit hinreichend großem R>0L; die Randbedingungen auf ∂Ω R werden durch das asymptotische Verhalten der Lösungen des Ausgangsproblems motiviert. Das Approximationsproblem wird mit der Methode der finiten Elemente in der Raumvariablen und dem Differenzenverfahren in der Zeitvariablen gelöst. Darüber hinaus wird ein gekoppeltes numerisches Schema für das Cauchy-Problem vorgeschlagen, weiters werden die Existenz und die Eindeutigkeit der zugehörigen Näherungslösung bewiesen sowie Fehlerschranken angegeben.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams, R. A.: Sobolev Spaces. New York: Academic Press 1975.
Carlson, D. E.: Linear Thermoelasticity, Handbuch der PhysikVIa/2, pp. 297–345. Berlin: Springer-Verlag 1972.
Ciarlet, P. G.: The Finite Element Method for Elliptic Problems. North-Holland 1978.
Dafermos, C.M.: Contemporary Issues in the Dynamic Behavior of Continuous Media. Lefschetz Center for Dynamical Systems Report, Brown Univ. 1985.
Douglas, Jr. J., Dupont, T.: Galerkin methods for parabolic equations with nonlinear boundary conditions. Numer. Math.20, 213–237 (1973).
Dupont, T.:L 2-Estimates for Galerkin methods for second order hyperbolic equations, SIAM J. Numer. Anal.10, 880–889 (1973).
Jiang, S.: A finite element method for equations of one-dimensional nonlinear thermoelasticity, J. Comput. Appl. Math.21, 1–16 (1988).
Jiang, S.: Far field behavior of solutions to the equations of nonlinear 1-d-thermoelasticity. To appear in: Applicable Analysis.
Kawashima, S.: Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. Thesis, Kyoto Univ. 1983.
Lees, M.: A priori estimates for the solutions of difference approximations parabolic differential equations. Duke. Math. J.27, 297–311 (1960).
Moura, C. A. De: A linear uncoupling numerical scheme for the nonlinear coupled thermoelastodynamics equations. In: Lecture Notes in Math. 1005. (Pereyra, V., Reinoze, A. ed.), pp. 204–211. Berlin: Springer 1983.
Moura, C. A. De, Suaiden, T.: Thermoelastodinâmica não-linear: experimêntos numéricos em torno do modelo unidimensional de Slemrod, Laboratório de Computação Cientifica, No.006/83.
Slemrod, M.: Global existence, uniqueness and asymptotic stability of classical smooth solutions in one-dimensional non-linear thermoelasticity. Arch. Rat. Mech. Anal.76, 97–133 (1981).
Vol'pert, A. I., Hudjaev, S. I.: On the Cauchy problem for composite systems of nonlinear differential equations. Math. USSR Sbornik16, 517–544 (1972).
Wheeler, M. F.: A prioriL 2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10, 723–759 (1973).
Zheng, S., Shen, W.: Global solutions to the Cauchy problem of quasilinear hyperbolic parabolic coupled systems. Sci. Sinica, Ser. A30, 1133–1149 (1987).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jiang, S. Numerical solution for the Cauchy problem in nonlinear 1-D-thermoelasticity. Computing 44, 147–158 (1990). https://doi.org/10.1007/BF02241864
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02241864