Adaptive Conservative Numerical Method of Sine-Gordon Equation Based on Multi-quadric Quasi-interpolation

Existing numerical methods are weak in simulating the SG（Sine-Gordon） equation for long time. They may also be time-consuming and inefficient when the solutions involve large variations. This paper proposes an adaptive and energy conservative approach based on the MQ（multi-quadric） to overcome these limits. Firstly, the MQ quasi-interpolations with symmetric kernels are employed to approximate the spatial derivatives of each variable; secondly the new knots on the next time step are obtained according to the moving knots equation; thirdly the Staggered St？rmer Verlet scheme is employed to approximate the temporal derivatives of each variable. The energy conservation estimation and the truncation error of the proposed scheme are presented. Numerical experiments demonstrate that the proposed method is easy to implement, accurate and able to simulate the SG equation for long time.