Numerical Method and Analysis for the Coupled Sine‐Gordon Equations on Unbounded Domains

ABSTRACT In this article, we delve into the efficient artificial boundary method, specifically focusing on its application in solving the coupled sine‐Gordon equations on unbounded domains, which are widely used in a variety of scientific fields. By incorporating the operator splitting approach, we have designed local artificial boundary conditions that effectively address the challenges posed by the unbounded nature of the physical domain and the intricate nonlinearities involved. These conditions are specifically developed for the coupled sine‐Gordon equations, ensuring that wave outgoing boundaries are achieved without any reflections. An initial boundary value problem on a bounded computational domain is obtained, and the finite difference method is adopted to discretize the reduced problem. Rigorous analysis of the stability and convergence of the reduced problem is conducted through the introduction of an energy function. Some numerical examples are presented to demonstrate the accuracy and effectiveness of the method.