Numerical Solution for Sine-Gordon System in One Dimension

This paper has studied the numerical solution for Sine-Gordon system in one dimensions using finite difference methods. We have used Explicit method and Crank-Nicholson method.A comparison between results of the two methods has been done and we obtained that Crank-Nicholson method is more accurate than the Explicit method but the Explicit method is easer .
We also studied the stability analysis for each method by using Fourier(Von-Neumann) method and obtained that Crank-Nicholson method is unconditionally stable while the Explicit method is stable under the condition and .