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 $0<\varepsilon\ll1$ 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.