Keywords : explicit method

Numerical Solution for Sine-Gordon System in One Dimension

Saad A. Manna; Haneen T. Jassim

AL-Rafidain Journal of Computer Sciences and Mathematics, 2010, Volume 7, Issue 2, Pages 47-59
DOI: 10.33899/csmj.2010.163896

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  .

The Finite Difference Methods for φ4 Klein-Gordon Equation

Saad A. Manaa; Laya Y. Hawell

AL-Rafidain Journal of Computer Sciences and Mathematics, 2008, Volume 5, Issue 2, Pages 47-63
DOI: 10.33899/csmj.2008.163970

We solved φ4 Klein-Gordon equation numerically by using two finite difference methods: The first is the explicit method and the second is the implicit (Crank-Nicholson) method. Also, we studied the numerical stability of the two methods using Fourier (Von Neumann) method and it has been found that the first method is simpler and has faster convergence while the second method is more accurate, and the explicit method is conditionally stable while the implicit method is unconditionally stable.