Numerical analysis of a finite volume method for a 1-D wave equation with non smooth wave speed and localized Kelvin-Voigt damping

TL;DR

This paper develops and analyzes a finite volume method for simulating 1-D wave equations with irregular wave speeds and localized damping, providing stability, convergence, and decay rate results validated by numerical experiments.

Contribution

It introduces a finite volume scheme for complex wave equations with non-smooth coefficients and localized damping, with proven stability and convergence properties.

Findings

The numerical method is stable under certain conditions.

Convergence of the scheme to the continuous solution is established.

Numerical experiments confirm the decay rate of the solution with localized damping.

Abstract

In this paper, we study the numerical solution of an elastic/viscoelastic wave equation with non smooth wave speed and internal localized distributed Kelvin-Voigt damping acting faraway from the boundary. Our method is based on the Finite Volume Method (FVM) and we are interested in deriving the stability estimates and the convergence of the numerical solution to the continuous one. Numerical experiments are performed to confirm the theoretical study on the decay rate of the solution to the null one when a localized damping acts.