Two regularized energy-preserving finite difference methods for the logarithmic Klein-Gordon equation

TL;DR

This paper introduces two energy-preserving finite difference methods for the logarithmic Klein-Gordon equation, employing a regularized version to handle singularities and providing error bounds and numerical validation.

Contribution

The paper proposes two novel regularized finite difference schemes that preserve energy for the LogKGE and analyzes their convergence with error bounds.

Findings

Error bound of O(h^2 + τ^2/ε^2) for the schemes

Regularized LogKGE approximates LogKGE with order O(ε)

Numerical results confirm theoretical analysis

Abstract

We present and analyze two regularized finite difference methods which preserve energy of the logarithmic Klein-Gordon equation (LogKGE). In order to avoid singularity caused by the logarithmic nonlinearity of the LogKGE, we propose a regularized logarithmic Klein-Gordon equation (RLogKGE) with a small regulation parameter $0<\varepsilon\ll1$ to approximate the LogKGE with the convergence order $O(\varepsilon)$. By adopting the energy method, the inverse inequality, and the cut-off technique of the nonlinearity to bound the numerical solution, the error bound $O(h^{2}+\frac{\tau^{2}}{\varepsilon^{2}})$ of the two schemes with the mesh size $h$, the time step $\tau$ and the parameter $\varepsilon$. Numerical results are reported to support our conclusions.