An implicit-explicit time discretization scheme for second-order semilinear wave equations with a nonlocal material law and kinetic boundary conditions

In this paper we construct and analyze an implicit-explicit (IMEX) scheme for the semilinear viscoacoustic wave equation with a retarded material law. It contains a convolution term with exponential kernels and is thus nonlocal in time. Furthermore, the wave equation is equipped with kinetic boundary conditions. We treat the convolution term via additional variables. In order to make the problem well-posed, it is essential to perform an appropriate shift to derive auxiliary differential equations which are coupled to the first–order formulation of the wave equation. For the kinetic boundary conditions, we consider these equations in weighted bulk-surface Sobolev spaces. Second–order error bounds in time are proven for the IMEX scheme and are supported by numerical experiments, where the IMEX scheme is combined with an isoparametric finite element discretization in space.