Numerical solution and stability analysis of the Sine-Gordon system in two dimensions

This paper deals with the numerical solution for Sine-Gordon system in two dimensions using two finite difference methods the (ADE) and (ADI) methods .A comparison between the two methods has been done and we have obtained that the (ADE) method is the easer while the (ADI) method is more accurate than the (ADE) method. We also studied the stability analysis for each method by using Fourier (Von Neumann) method and we have obtained that the (ADI) method is unconditionally stable while the (ADE) method is stable under the condition and where is the ratio of the waves speed u , w and .