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

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 $0<\epsilon \ll 1$ to approximate the LogKGE with the convergence order $O(\epsilon )$. By adopting the energy method, the inverse inequality, and the cut-off technique of the nonlinearity to bound the numerical solution, the error bound $O(h^2+\frac{{\tau }^2}{{\epsilon }^2})$ of the two schemes with the mesh size $h$, the time step $\tau$ and the parameter $\epsilon$. Numerical results are reported to support our conclusions.

arXiv commentarXiv admin note: text overlap with arXiv:2006.08079

References in corpus (5)

• Q-Balls and Baryogenesis in the MSSM
• Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method
• Geometric properties of Kahan's method
• Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime
• Uniform error bounds of a time-splitting spectral method for the long-time dynamics of the nonlinear Klein-Gordon equation with weak nonlinearity