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Uniform stabilization of a viscous numerical approximation for a locally damped wave equation. (English) Zbl 1120.65101
The paper addresses the analysis of discretizations of a locally damped wave equation in a regular two-dimensional domain. The damping is chosen such that the energy of the exact solutions decays exponentially in time. For space (semi-)discretization, 5-point finite differences of order two are employed, where a non-standard viscous term is added to dampen spurious numerical oscillations. It is shown that also for this semi-discrete equation, the energy decays exponentially, uniformly in the mesh size. Subsequently, convergence of the semi-discretization to the solution of the continuous problem is proven in a weak sense. Finally, numerical experiments indicate convergence of the full discretization as well.

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35L15 Initial value problems for second-order hyperbolic equations
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