Regularized finite difference methods for the logarithmic Klein-Gordon equation

TL;DR

This paper develops and analyzes two regularized finite difference methods for the logarithmic Klein-Gordon equation, addressing blowup issues by introducing a small regularization parameter and providing error bounds and convergence analysis.

Contribution

It introduces a regularized version of the LogKGE and proposes two finite difference schemes with rigorous error analysis and convergence results.

Findings

Error bounds depend on mesh size, time step, and regularization parameter

Numerical experiments confirm theoretical error estimates

Convergence to the original LogKGE with linear order in regularization parameter

Abstract

We propose and analyze two regularized finite difference methods for the logarithmic Klein-Gordon equation (LogKGE). Due to the blowup phenomena caused by the logarithmic nonlinearity of the LogKGE, it is difficult to construct numerical schemes and establish their error bounds. In order to avoid singularity, we present a regularized logarithmic Klein-Gordon equation (RLogKGE) with a small regularized parameter $0<\varepsilon\ll1$. Besides, two finite difference methods are adopted to solve the regularized logarithmic Klein-Gordon equation (RLogKGE) and rigorous error bounds are estimated in terms of the mesh size $h$, time step $\tau$, and the small regularized parameter $\varepsilon$. Finally, numerical experiments are carried out to verify our error estimates of the two numerical methods and the convergence results from the LogKGE to the RLogKGE with the linear convergence order $O(\varepsilon)$.