Fourth-order energy-preserving locally implicit time discretization for linear wave equations

A family of fourth-order coupled implicit-explicit time schemes is presented as a special case of fourth-order coupled implicit schemes for linear wave equations. The domain of interest is decomposed into several regions where different fourth-order time discretizations are used, chosen among a family of implicit or explicit fourth-order schemes. The coupling is based on a Lagrangian formulation on the boundaries between the several non-conforming meshes of the regions. A global discrete energy is shown to be preserved and leads to global fourth-order consistency in time. Numerical results in 1D and 2D for the acoustic and elastodynamics equations illustrate the good behavior of the schemes and their potential for the simulation of realistic highly heterogeneous media or strongly refined geometries, for which using everywhere an explicit scheme can be extremely penalizing. Accuracy up to fourth order reduces the numerical dispersion inherent to implicit methods used with a large time step and makes this family of schemes attractive compared with second-order accurate methods.