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Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity. (English) Zbl 1033.65080
The authors consider the finite-difference semi-discretization in space of a locally damped wave equation in an interval and on a square. The damping is supported in a suitable subset of the domain under consideration so that the energy of the solutions decays exponentially to zero as time goes to infinity. The decay rate of the semi-discrete systems depends on the mesh size $h$ of the discretization and tends to zero as $h$ goes to zero. It is shown that adding a suitable vanishing numerical viscosity term leads to an $h$-uniform exponential decay of the energy of the solutions by damping out the spurious high frequency numerical oscillations while the convergence of the scheme towards the solutions of the original damped wave equation, as $t\rightarrow \infty$, is maintained in a suitable topology. The suitability of numerical damping is closely connected to the efficiency of the Tikhonov regularization techniques.

MSC:
 65M20 Method of lines (IVP of PDE) 65M06 Finite difference methods (IVP of PDE) 35L15 Second order hyperbolic equations, initial value problems 65M12 Stability and convergence of numerical methods (IVP of PDE)
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