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. In many physical problems where accurate dynamic models are required, the contribution of some underlying and more or less ill-known distributed phenomena cannot be neglected. Although the precise local description of such phenomena often appears excessively complex or even, in many cases, out of scope, their macroscopic dynamic consequences can fortunately most of time be taken into account by means of suitable time-operators of convolution nature which in fact summarize the collective contribution of lots of hidden parameters to the global dynamic behavior of quantities under interest. In that sense, such integrodifferential models therefore conciliate accuracy and simplicity, up to the loss of the so-called time-locality property: in opposite to standard Cauchy problems for which the future is conditioned by the present only, all the past evolution is involved here, via the time-convolution. Last years, various problems relating to integrodifferential models have been studied in many fields. As few examples, we can cite [2,7,13,16] in physics, [6,10,12] in mathematical analysis or numerical simulation, [1,11] in control problems, [3,9] in electrical engineering, [18] in biology, etc.In the particular context of partial integrodifferential equations, the crucial problem of numerical simulation is in general quite difficult. This is due for one part to the numerical complexity of quadratures of convolution integrals, which generate highly expensive time discretizations, particularly when long memory components are present. Beyond this first heavy shortcoming, the stability of numerical schemes is in general very difficult to get, namely because standard techniques devoted to (ordinary) partial differential equations such as energy dissipation cannot be used for integrodifferential equations. So, the construction of stable numerical schemes remains an important challenge and it can be expected that some specific methods devoted to analysis a...
