Regularized finite difference methods for the logarithmic Klein-Gordon equation

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 $O(\varepsilon)$.