A novel regularized model for the logarithmic Klein-Gordon equation

Abstract

In this paper, the energy regularization technique proposed in [Bao et al., arXiv:2006.05114] is adopted for the logarithmic Klein-Gordon equation to avoid the singularity of the logarithmic function at the origin. The convergence rate between the energy functions of the energy regularized logarithmic Klein-Gordon equation (ERLogKGE) and the logarithmic Klein-Gordon equation (LogKGE) is proved to be O(ε2). An energy preserving Crank-Nicolson finite difference method and an explicit finite difference method are presented for the ERLogKGE. By adopting the mathematical induction, energy method, and inverse inequality, the error bound of the explicit finite difference method is estimated in terms of mesh size h, time step τ, and the regularization parameter ε. For comparison reasons, two regularization techniques for the logarithmic nonlinearity are also studied for the LogKGE. A collection of numerical experiments show that the proposed energy regularization technique is superior to other regularized schemes.