The numerical solution of the nonlinear Klein–Gordon and Sine–Gordon equations using the Chebyshev tau meshless method

In this work, we study the numerical solutions of one-dimensional Klein–Gordon and Sine–Gordon equations using the Chebyshev tau meshless method based on the integration–differentiation (CTMMID). First, we apply CTMMID to discretize both space and time variables. The initial and boundary conditions could be incorporated efficiently with full CTMMID. Furthermore, we introduce the Domain Decomposition Method (DDM) in space and the block-marching technique in time for problems defined in large interval and long time computing. The numerical results are more accurate and with less computational effort than some existing studies.