A pseudo-spectral method that uses an overlapping multidomain technique for the numerical solution of sine-Gordon equation in one and two spatial dimensions

摘要

In this article,we study an explicit scheme for the solution of sine-Gordon equationwhen the space discretization is carried out by an overlapping multidomain pseudo-spectral technique. By using differentiation matrices, the equation is reduced to a nonlinear system of ordinary differential equations in time that can be discretized with the explicit fourth-order Runge–Kutta method. To achieve approximation with high accuracy in large domains, the number of space grid points must be large enough. This yields very large and full matrices in the pseudo-spectral method that causes large memory requirements. The domain decomposition approach provides sparsity in the matrices obtained after the discretization, and this property reduces storage for large matrices and provides economical ways of performing matrix–vector multiplications. Therefore, we propose a multidomain pseudo-spectral method for the numerical simulation of the sine- Gordon equation in large domains. Test examples are given to demonstrate the accuracy and capability of the proposed method. Numerical experiments showthat the multidomain scheme has an excellent long-time numerical behavior for the sine-Gordon equation in one and two dimensions.