Numerical schemes for the solution of the damped wave equation
Résumé
The present work deals with the numerical resolution of the damped wave equation. Thus, we present various numerical explicit and implicit schemes for problems defined in one-, two-and three-dimensional domain. In the case of one-dimensional problem, we consider, on one hand, Dirichlet boundary value conditions and in the other hand, mixed boundary value conditions. In the two-and three-dimensional cases, we consider only the case of Dirichlet boundary value conditions. However, in all cases, we analyze the stability and the truncation error of the presented schemes. Concerning the implicit schemes, we survey the main classical numerical linear algebra algorithms and show that they can be applied successfully to the numerical solution of the target problem, in the two-and three-dimensional cases on sequential or multiprocessor computers. We briefly consider also the cases where, on the one hand, the coefficients of the problem are not constant and, on the other hand, the case where the linear problem is perturbed by a singlevalued or a multivalued diagonal operator and we explain how to solve implicitly this type of problem on a multiprocessor. This study is completed by numerical simulations in the case of one-dimensional problems. Finally, to study the stability of the schemes, we can either use the classical Von Neumann method or the matricial one when the discretization matrix in space is normal. In this case, it is necessary to have the eigenvalues of the space discretization matrices. Thus, an appendix presents the determinination of these eigenelements in academic situations.
Origine | Fichiers produits par l'(les) auteur(s) |
---|