Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity
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- Tébou, L. & Zuazua, E. Numer. Math. (2003) 95: 563. doi:10.1007/s00211-002-0442-9
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We consider the finite-difference space semi-discretization of a locally damped wave equation, the damping being supported in a suitable subset of the domain under consideration, so that the energy of solutions of the damped wave equation decays exponentially to zero as time goes to infinity. The decay rate of the semi-discrete systems turns out to depend on the mesh size h of the discretization and tends to zero as h goes to zero. We prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size) exponential decay of the energy of solutions. This numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept. We discuss this problem in 1D and 2D in the interval and the square respectively. Our method of proof relies on discrete multiplier techniques.