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

Stefano Giani,Luka Grubišić,Christian Engström

doi:10.1016/j.aml.2021.107844

Stefano Giani, Luka Grubišić + Show 1 more

Open Access

https://doi.org/10.1016/j.aml.2021.107844

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Abstract

In this paper, we compare two approaches to numerically approximate the solution of second-order Gurtin–Pipkin type of integro-differential 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 efficient 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.

Highlights

IntroductionWhich is based on a spectral decomposition of the positive self-adjoint operator A and the inverse Laplace transform
We propose a numerical method for the wave equation
The projection-based approach described above is numerically compared with a standard inverse Laplace transform technique, which requires that many source problems be solved for each time point

Summary

Introduction

Which is based on a spectral decomposition of the positive self-adjoint operator A and the inverse Laplace transform. The spectrum of T contains branches of eigenvalues with unbounded imaginary parts but the projected operator function TN will only have spectral points with finite imaginary parts. This makes it straightforward to deform the contour in the Bromwich inversion formula into C− := {λ ∈ C : Re λ < 0}, which is Talbot’s approach [9]. The projection-based approach described above is numerically compared with a standard inverse Laplace transform technique, which requires that many source problems be solved for each time point.

The spectral projection based method

The standard approach

Numerical results

Conclusions