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

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.