Time-local dissipative formulation and stable numerical schemes for a class of integrodifferential wave equations

Céline Casenave (1) , Emmanuel Montseny (2)

1 LAAS-MRS
2 LAAS-DISCO - Équipe DIagnostic, Supervision et COnduite

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.

Mots clés: integrodifferential equation; partial differential equation; convolution operator; diffusive representation; numerical scheme; Cauchy problem; energy functional; stability condition

Domaines: Automatique / Robotique; Analyse numérique [math.NA]

Céline Casenave, Emmanuel Montseny. Time-local dissipative formulation and stable numerical schemes for a class of integrodifferential wave equations. SIAM Journal on Applied Mathematics, 2008, 68 (6), pp.1763--1783. ⟨10.1137/070693710⟩. ⟨hal-00831402⟩