Numerical approximations for a hyperbolic integrodifferential equation with a non-positive variable-sign kernel and nonlinear-nonlocal damping

TL;DR

This paper develops and analyzes Galerkin-based numerical schemes for a hyperbolic integrodifferential equation with a non-positive kernel and nonlinear damping, establishing stability, uniqueness, and error estimates verified by numerical experiments.

Contribution

It introduces a fully discrete Galerkin scheme for the equation with stability and error analysis, including novel semi-norms, and confirms results through numerical tests.

Findings

Long-time stability of solutions established

Finite-time error estimates derived for semi-discrete and fully discrete schemes

Numerical experiments verify theoretical stability and error bounds

Abstract

This work considers the Galerkin approximation and analysis for a hyperbolic integrodifferential equation, where the non-positive variable-sign kernel and nonlinear-nonlocal damping with both the weak and viscous damping effects are involved. We derive the long-time stability of the solution and its finite-time uniqueness. For the semi-discrete-in-space Galerkin scheme, we derive the long-time stability of the semi-discrete numerical solution and its finite-time error estimate by technical splitting of intricate terms. Then we further apply the centering difference method and the interpolating quadrature to construct a fully discrete Galerkin scheme and prove the long-time stability of the numerical solution and its finite-time error estimate by designing a new semi-norm. Numerical experiments are performed to verify the theoretical findings.