Keywords : Von-Neumann Method

Using Exponential Finite Difference Method for Solve Kuramoto-Sivashinsky Equation with Numerical Stability Analysis

Abdulghafor M. Al-Rozbayani; Shrooq M. Azzo

AL-Rafidain Journal of Computer Sciences and Mathematics, 2013, Volume 10, Issue 4, Pages 147-165
DOI: 10.33899/csmj.2013.163562

In this paper we solved the Kuramoto-Sivashinsky Equation numerically by finite-difference methods, using two different schemes which are the Fully Implicit scheme and Exponential finite difference scheme, because of the existence of the fourth derivative in the equation we suggested a treatment for the numerical solution of the two previous scheme by parting the mesh grid into five regions, the first region represents the first boundary condition, the second at the grid point x1, while the third represents the grid points x2,x3,…xn-2, the fourth represents the grid point xn-1and the fifth is the second boundary condition. We also, studied the numerical stability by Fourier (Von-Neumann) method for the two scheme which used in the solution on all mesh points to ensure the stability of the point which had been treated in the suggested style, we using two interval with two initial condition and the numerical results obtained by using these schemes are compare with Exact Solution of Equation Excellent approximate is found between the Exact Solution and numerical Solutions of these methods.

Alternating Direction Implicit Method for Solving Parabolic Partial Differential Equations in Three Dimensions

Abdulghafor M. Al-Rozbayani; Mahmood H. Yahya

AL-Rafidain Journal of Computer Sciences and Mathematics, 2012, Volume 9, Issue 2, Pages 79-97
DOI: 10.33899/csmj.2012.163703

In this paper, the parabolic partial differential equations in three-dimensions are solved by two types of finite differences, such as, Alternating Direction Explicit (ADE) method and Alternating Direction Implicit (ADI) method. By the comparison of the numerical results for the previous two methods with the Exact solution, we observe that the results of Alternating Direction Implicit (ADI) method is better and nearest to the exact solution compared with the results of Alternating Direction Explicit (ADE) method. we also studied the numerical stability of both methods by Von-Neumann Method.