Time-Local Dissipative Formulation and Stable Numerical Schemes for a Class of Integrodifferential Wave Equations

Abstract

We consider integrodifferential equations of the abstract form $\mathbf{H} (\partial_{t})\Phi=\mathbf{G}(\nabla)\Phi+f$, where $\mathbf{H}(\partial_{t})$ is a diagonal convolution operator and $\mathbf{G}(\nabla)$ 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 $\partial_{t}\Psi= \mathcal{A}\Psi+\mathcal{B}f$ such that $\Phi=\mathcal{C}\Psi$. We show that under a suitable hypothesis on the symbol $\mathbf{H}(p)$, 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.