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

79 Alternating Direction Implicit Method for Solving Parabolic Partial Differential Equations in Three Dimensions Abdulghafor M. Al-Rozbayani Mahmood H. Yahya abdulghafor_rozbayani@uomosul.edu.iq College of Computer Sciences and Mathematics, University of Mosul, Iraq Received on: 2/05/2011 Accepted on: 16/08/2011 ABSTRACT 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.


Introduction:
Partial differential equations (PDEs) form the basis of very many mathematical models of physical, chemical and biological phenomena, and more recently they spread into economics, financial forecasting, image processing and other fields. To investigate the predictions of PDE models of such phenomena, it is often necessary to approximate their solution numerically, commonly in combination with the analysis of simple special cases; while in some of the recent instances the numerical models play an almost independent role [10].
Parabolic partial differential equations in two or three space dimensions with over-specified boundary data feature in the mathematical modeling of many important phenomena. While a significant body of knowledge about the theory and numerical methods for parabolic partial differential equations with classical boundary conditions has been accumulated, not much has been extended to parabolic partial differential equations with over-specified boundary data [4]. We often meet the problem of solving equation of parabolic type in many fields such as seepage, diffusion, heat conduction and so on [9].
B.J. Noye and K.J. Hayman in [11] used ADI to solve the two dimensional timedependent heat equations subject to a constant coefficient, J.M. McDonough in [12] used ADI methods for solving elliptic problems and Norma Alias and Md. Rajibul Islam in [1] used alternating group explicit (AGE) method and Iterative alternating decomposition explicit (IADE) method to solve a two-dimensional and threedimensional in PDE problems. Mohamed A. Antar and Esmail M. Mokheimer in [2] used spreadsheet programs to solve a three dimensional equation for numerical solutions by using finite difference solutions which are the most appropriate.
In this paper, we study and apply the finite difference methods to approximate the solution and study the stability of the numerical solution of a model of parabolic partial differential equation in three dimensions. These methods are combinations of finite difference method with -Alternating direction explicit method (ADE) -Alternating direction implicit method (ADI) First, we derive the finite differential form of ADE and ADI methods for the given model and then present an algorithm for each method. Also we compare between them. The stability for the above methods has been examined .

Numerical Methods
We solve the mathematical model in (1) with the combination of the finite difference methods with ADE and ADI methods.

ADE Method
The alternating direction explicit (ADE) method for generating numerical solutions to the diffusion equation is stable for some time because it is an explicit method; it holds a speed advantage over implicit methods for computations over a single time level [7] the explicit methods in which the solution at the new time step is formed by a combination of pervious time step solutions [13,14].
When we consider a square region ( ) and that u is known at all points within and on the boundary of the square region; we draw lines parallel to x , y , z , taxis as

ADI Method
The ADI was first suggested by Douglas , Peaceman and Rachford [3,4,11] for solving the heat equation in two spatial variables and alternating direction implicit (ADI) methods have proved valuable in the approximation of the solutions of parabolic and elliptic differential equations in two and three variables [6,7] In the ADI approach, the finite difference equations are written in terms of quantities at three x levels. However, three different finite difference approximations are used alternately, one to advance the calculations from the plane n to a plane (n+1), the second to advance the calculations from (n+1) plane to the (n+2) plane and the third to advance the calculations from (n+2) plane to the (n+3) plane [10]. Then, we advance the solution of the parabolic partial differential equation in three dimensions from nth plane to (n+1)th plane by replacing      u  u  u  r  u  u  u  u  r  u   1  ,  ,  ,  ,  1  ,  ,   ,  1  ,  ,  ,  ,  1  ,  ,  ,   1  ,  ,  1 1 , , Also, we advance the solution from the (n+1)th plane to (n+2)th plane by replacing 2 2 y u   by implicit finite difference approximation at (n+2)th plane. Similarly, then, we have a square region and multiply eq.(10) by t  then we get We simplify and rearrange the above equation and we get Now, we advance the solution from (n+2)th plane to (n+3)th plane by replacing We simplify and rearrange the above equation and we get    ,  ,  3  1  ,  ,  3  ,  1  ,  3  ,  1  ,  3  ,  ,  3   1  ,  ,  2  1  ,  ,  2  ,  1  ,  2  ,  1

Numerical Stability
There two methods, we used here one including the effect of boundary are conditions and the other excluding the effect of boundary conditions which are used to investigate stability. Both methods are attributed to John von Neumann. These approaches are Fourier and matrix methods. Fourier method, the primary observation in the Fourier method is that the numerical scheme is linear and therefore it will have solution in the form ( )

Stability Analysis of ADE Method
The Von-Neumann method has been used to study the stability analysis of Parabolic model in three dimensions.
We can apply this method by substituting the solution in finite difference method at the time t by Considering the left-side inequality (as the right-side inequality is always true), We have Thus the ADE method for eq.(1) is conditionally stable.

Stability Analysis of ADI Method
The ADI finite difference method from eq. And substitute (15) in (9) By using Euler formula   By using Euler formula as previously Where   ,   and   stand for the  plane,  plane and  plane. However, in either form unconditional stability is lost. Furthermore, the combined three-levels have the form : ; this shows that the ADI method is conditionally stable in three-dimensional problem. Therefore, the combined three-levels are conditionally stable [6,17].

Example (1) [9]:
We consider the initial and boundary value problem as follows : By using the numerical methods such as ADE method and ADI method of (21), we take the parameters 10 for convenience using the exact solution of (21) ( . Also, we compute the stability of each of the above methods and we conclude that the ADE method is conditionally stable where, 6 1  r and ADI methods are also conditionally stable where, 2 1  r is compared between them and with the exact solution.

Example (2) [16]:
( ) We solve the following initial-boundary value problem : We use the numerical methods such as ADE method and ADI methods of (22).
We take the parameters 10 for convenience by using the exact solution of (22) ( . Also, we compute the stability of each of the above methods and we conclude that the ADE method is conditionally stable where, 6 1  r and ADI methods are also conditionally stable where, 2 1  r is compared between them and with the exact solution.

Conclusion
Through our study for numerical stability to the ADE method for PDEs in threedimensional, we conclude that it is conditionally stable such as in two dimensions equations, but the ADI method is lost the unconditionally stable that is in two dimensions. Also, we saw that from the numerical results the ADI method is better than the ADE method and its results are nearest to the exact solution compared with the results of ADE method.