Keywords : Crank-Nicholson scheme


Numerical Solution and Stability Analysis for Burger's-Huxley Equation

Saad A. Manaa; Farhad M. Saleem

AL-Rafidain Journal of Computer Sciences and Mathematics, 2009, Volume 6, Issue 2, Pages 61-75
DOI: 10.33899/csmj.2009.163797

The Burger’s-Huxley equation has been solved numerically by using two finite difference methods, the explicit scheme and the Crank-Nicholson scheme. A comparison between the two schemes has been made and it has been found that, the first scheme is simpler while the second scheme is more accurate and has faster convergent. Also, the stability analysis of the two methods by using Fourier (Von Neumann) method has been done and the results were that, the explicit scheme is stable under the condition   and the Crank-Nicholson is unconditionally stable.
 

The Finite Difference Methods for Hyperbolic – Parabolic Equations

Abbas Y. Al-Bayati; Saad A. Manaa; Ekhlass S. Al-Rawi

AL-Rafidain Journal of Computer Sciences and Mathematics, 2005, Volume 2, Issue 2, Pages 57-71
DOI: 10.33899/csmj.2005.164084

The objective of this paper is to construct numerical schemes using finite difference methods for the one-dimensional general hyperbolic- parabolic- reaction problem.
The finite difference method with the exponential transformation form is used to solve the problem, and employs difference approximation technique to obtain the numerical solutions. Computational examples are presented and compared with the exact solutions. We obtained that the Crank-Nicholson scheme is more accurate than Forward scheme. Therefore the form of exponential transformation for the problem yields a stable solution compared with exact solution.