Time-local dissipative formulation and stable numerical schemes for a class of integrodifferential wave equations
Résumé
We consider integrodifferential equations of the abstract form H(∂t)Φ=G(∇)Φ+f, where H(∂t) is a diagonal convolution operator and G(∇) is a linear anti-selfadjoint differential operator. On the basis of an original approach devoted to integral causal operators, we propose and study a time-local augmented formulation under the form of a Cauchy problem ∂tΨ=AΨ+Bf such that Φ=CΨ. We show that under a suitable hypothesis on the symbol H(p), this new formulation is dissipative in the sense of a natural energy functional. We then establish the stability of numerical schemes built from this time-local formulation, thanks to the dissipation of appropriate discrete energies. Finally, the efficiency of these schemes is highlighted by concrete numerical results relating to a model recently proposed for 1D acoustic waves in porous media.
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Preprint_-_Casenave_and_Montseny_-_2008_-_Time-local_dissipative_formulation_and_stable_nume.pdf (288.36 Ko)
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