Abstract
In this paper, the object of our study is two coupled partial differential equation which have a rich physical background. With a reasonable change to the initial value, we use central difference method and compact finite difference method to get the numerical results of the governing equations. In the analysis of the latter, we use the Hermitian and skew-Hermitian splitting (HSS) method to solve a large sparse non-Hermitian positive definite system of linear equations. In this way, the computational efficiency can be improved and the convergence is guaranteed which is proved by Bai [Bai et al. [2003] "Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems," SIAM J. Matrix Anal. Appl. 24, 603-626] and Chen [Chen et al. [2014] "Convergence analysis of the modified Newton-HSS method under the Holder continuous condition," J. Comput. Appl. Math. 264, 115-130]. And we also pay attention to the effect of initial conditions, on which we add a small perturbation to observe its influence by comparing numerical results.