A spectral projection based method for the numerical solution of wave equations with memory

Abstract

In this paper, we compare two approaches to numerically approximate the solution of second-order Gurtin-Pipkin type of integro-dierential equations. Both methods are based on a high-order Discontinous Galerkin approximation in space and the numerical inverse Laplace transform. In the first approach, we use functional calculus and the inverse Laplace transform to represent the solution. The spectral projections are then numerically computed and the approximation of the solution of the time-dependent problem is given by a summation of terms that are the product of projections of the data and the inverse Laplace transform of scalar functions. The second approach is the standard inverse Laplace transform technique. We show that the approach based on spectral projections can be very ecient when several time points are computed, and it is particularly interesting for parameter-dependent problems where the data or the kernel depends on a parameter.