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

TL;DR

This work addresses numerical approximation for a hyperbolic integrodifferential equation with a non-positive variable-sign kernel and nonlinear-nonlocal damping. It introduces a kernel transform to a positive-type kernel via $K(t)=\int_t^{\infty}\beta(s)\,ds$, enabling energy-based well-posedness analysis that yields long-time stability and finite-time uniqueness for the continuous problem. The authors develop a spatial Galerkin semi-discrete scheme and a fully discrete scheme using centering differences and interpolating quadrature, establishing stability and $H^1$-error bounds with a novel semi-norm $\|\cdot\|_A$ and a careful handling of nonlinear-nonlocal terms; they also show an $O(h + \tau^{1+\alpha})$ temporal accuracy with $\alpha\in\{1,\tfrac{1}{2}\}$ depending on kernel regularity. Numerical experiments in 1D and 2D confirm the predicted convergence rates and demonstrate energy-dissipation behavior, validating the theoretical results. The approach advances reliable long-time simulations for viscoelastic-type models with memory and non-monotone kernels, and lays groundwork for future $L^2$-error analysis and energy-decay investigations.

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.

Figures (1)

• Figure 1: Plots of the energy under $f=0$, $\sigma=3$, $\gamma=3\sqrt{3}$, and $N=M=32$.

Theorems & Definitions (11)

• Lemma 2.1
• Theorem 2.2
• proof
• Theorem 3.1
• Theorem 3.2
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof