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

Abstract. 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 self-adjoint 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 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. Key words. integrodifferential equation, partial differential equation, convolution operator, diffusive representation, numerical scheme, Cauchy problem, energy functional, stability condition. AMS subject classifications. 1. Introduction. I