High Accuracy Difference Schemes For The Nonlinear Sine-Gordon Equation

In this paper, finite difference numerical methods for the nonlinear Sine-Gordon equation have been studied. Firstly, three difference schemes for the nonlinear Sine-Gordon equation have been given out. They are four-order three levels explicit scheme, four-order three levels implicit scheme, high-order three levels compact scheme. The analysis of the local truncation error of these three difference schemes has been gained respectively. The stability and convergence of these difference schemes have been analyzed. Their truncation errors and speeds have been compared by the numerical experiments; furthermore their effectiveness and reliability are demonstrated. On this basis, the nonlinear Sine-Gordon equation has been extended to the high-dimension nonlinear Sine-Gordon equation and the generalized nonlinear Sine-Gordon equation. Three difference schemes for the high-dimension nonlinear Sine-Gordon equation, and three difference schemes for the generalized nonlinear Sine-Gordon equation have been given out. The local truncation error, stability and convergence of these difference schemes have been analyzed respectively, and the numerical experiments are made to confirm their feasibility.