The Finite Difference Methods for Hyperbolic – Parabolic Equations

Abbas Y. Al-Bayati Saad A. Manaa Ekhlass S. Al-Rawi profabbasalbayati@yahoo.com drekhlass-alrawi@uomosul.edu.iq College of Computer Sciences and Mathematics University of Mosul/Iraq Received on: 12/04/2005 Accepted on: 30/05/2005 ABSTRACT The objective of this paper is to construct numerical schemes using finite difference methods for the one-dimensional general hyperbolicparabolicreaction 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.


1.Introduction:
Various finite difference algorithms or schemes have been presented for the solution of hyperbolic-parabolic problem or its simpler derivatives, such as the classical diffusion equation .It is well-known that many of these schemes are partially unsatisfactory due to the formation of oscillations and numerical diffusion within the solutions (Smith et al.,1973). Solution by the finite difference method, although more general, will involve stability and convergence problems, may require special handling of boundary conditions, and may require large computer storage and execution time. The problem of numerical dispersion for finite difference solutions is also difficult to overcome (Guymon,1970).
The hyperbolic-parabolic equation is a linear, nonsymmetrical partial differential equation of the parabolic type. The nonsymmetry arises from the so-called hyperbolic terms In this paper, the authors, using the exponential transformation, treat the classical differential representation for one-dimensional hyperbolicparabolic-reaction problems, and employ difference approximation techniques to obtain the numerical solutions. The comparison with the exact solutions shows good numerical approximations in the two examples.

2.The Mathematical Model:
The one-dimensional transport equation of a general hyperbolicparabolic-reaction problem of the form (see Chen et al. ,1998) is a linear function of  , the boundary conditions are (2) and the initial condition is Equation (1) with its auxiliary conditions is an approximate mathematical model of some physical problem. They make use of the fact that although (1) is nonsymmetrical, any second-order partial differential equation can be rendered symmetrical by the use of a transformation. In this case of (1), the transformation is simply (see Smith et al. ,1973) which after taking first and second derivatives and substituting into equation (1) yields the following transformed partial differential equation: The transformed geometric boundary condition becomes The transformed natural boundary condition is Finally, the initial condition transforms to

Numerical Solution by the Finite Difference Methods:
We consider the approximate solution of the problem and its transformation by using finite difference equations (FDE). That is we will use two methods: forward finite difference method (explicit method) and Crank-Nicholson method (implicit method) to solve these problems.
We introduce a uniform grid by defining the following discrete set of points in the x, t plane: In order to solve equations (1) and (5) numerically, we replace the differential equations with an analogous difference equations. Firstly, we apply two finite difference methods for solving equation (1), and secondly, we also apply these methods for solving equation (5).

Computational Examples:
In this section, we have solved two different problems and we have compared between them using the transformation in (4). We compare the results with the exact solutions by using MATLAB language.

Hyperbolic-parabolic problem:
The first test problem is chosen from Peaceman (1977). It is a onedimensional linear unsteady hyperbolic-parabolic problem described by The analytical solution of this problem is given by Karamouzis (1990 where erfc is the complementary error function, defined by:   Figures (1, 2, 3, 4) illustrate the numerical solutions of schemes (3.1) and (3.2) without and with transform of the problem (4.1), respectively. A comparison between the numerical approximations and the exact solution at t = 0.45 is given in figure (6) and table (1), which shows good numerical approximations by using exponential transformation for the problem. That is when we increase the value of time t we get a smooth curve by using exponential transformation method which means that it is more accurate than without this transform.

4.2Hyperbolic-parabolic-reaction problem:
The second test problem is an unsteady hyperbolic-parabolicreaction problem described by (see Chen et al. (1998) x e x  (4) which reveal that good results have been obtained. That is when we increase the value of time t we get a smooth curve by using exponential transformation method which means that it is more accurate than without this transform. And we obtain that the Crank-Nicholson scheme is more accurate than Forward scheme.

Conclusions:
The finite difference methods with the exponential transformation form are used to solve the classical differential representation for onedimensional hyperbolic-parabolic-reaction problems, and employed difference approximation techniques to obtain the numerical solutions.
Two different types of problems, hyperbolic-parabolic, and hyperbolic-parabolic-reaction, have been tested. We have compared between them using the exponential transformation form in eq.(4), we compared the results with the exact solutions which shows good numerical approximations in both examples. We have also 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 the exact solution. That is when we increase the value of time t we get a smooth curve by using exponential transformation method which means that it is more accurate than without this transform.