Numerical Solution and Stability Analysis of the Sine-Gordon Equation

The Sine Gordon equation has been solved numerically by using two finite differences methods: The first is the explicit scheme and the second is the Crank Nicholson scheme. A comparison between the two schemes has been made and the results were found to be : the first scheme is simpler and has faster convergence while the second scheme is more accurate . Also , the stability analysis of the two methods by the use of Fourier (Von Neumann) method has been done and the results were found to be : The explicit scheme is conditionally stable if and the CrankNicholson is unconditionally stable .