Description and analysis of schemes, fourth-order in time and space, for 1-D and 2-D acoustic equations

The purpose of this paper is the construction and the analysis of fourth order finite difference schemes for the numerical solution of acoustic wave propagation in fully heterogeneous media, i.e. with variable density and mobility. We present the main tools and results on the 1D model problem. In the case of a homogeneous medium, the fourth order space approximation is obtained by a classical five points finite difference approach and the fourth order explicit time discretization is handled using three time steps by the modified equation method. The extension of this scheme is conceived in order to obtain a variational formulation of the second order, which ensures the conservation of a certain discrete energy. We show that if one does not treat discontinuous coefficients with care enough, one may generates interface instabilities, illustrated by numerical examples. As a complement, an analysis of the approximation of transmission and reflection phenomena at an interface is given. The method can be extended to the 2D wave equation. He first show how to construct a family of explicit fourth-order in space ant time approximations of the d'Alembertian operator using three consecutive time steps and 25 points in space. These approximations depend on 2 parameters. We compare these schemes through an analysis of their dispersion and anisetropy properties. We extract four particularly interesting schemes that we extend to heterogeneous media by generalizing the ideas used for the 1D case. Results of numerical simulations are provided to illustrate the theoretical results.