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 << 1 to approximate the LogKGE with the convergence order O(epsilon). To derive the error bound of the two schemes, we combine the energy method, the inverse inequality, with the cut-off technique of the nonlinearity. And we obtain the error estimate at O(h(2) + tau(2)/epsilon(2)) for the two schemes with the mesh size h, the time step tau and the parameter epsilon. Numerical results are reported to support our conclusions. (c) 2021 Elsevier B.V. All rights reserved.

Automated summary

The paper introduces two regularized finite difference methods that preserve the energy of the logarithmic Klein-Gordon equation, and proposes a regularized logarithmic Klein-Gordon equation with a small regulation parameter to approximate the LogKGE. By combining the energy method, inverse inequality, and cut-off technique of the nonlinearity, the error bound of the two schemes is derived with an error estimate reported for the mesh size, time step, and parameter. Numerical results are provided to support the conclusions.