Explicit- Implicit Runge-Kutta Methods to Solve the System of Nonlinear PPDEs in Two Dimensions

جذومنل ةيددعلا لولحلا داجيإ مت ثحبلا اذه يف Gray-Scott وه جذومنلا اذه ،يناثلا دعبلا ءاضف يف ىـلإ يلصلأا جذومنلا ليوحت مت ثيح يطخ ريغلا ئفاكملا عطقلا عون نم ةيئزجلا ةيلضافتلا تلاداعملا نم ماظن عيطقتلا هبش مادختساب ةيدايتعلاا ةيلضافتلا تلاداعملا نم مظن تا ) MOLs .( جـنار قئارط انلمعتسا مث اـتوك ) جنار ةقيرط جنار ةقيرطو ةحيرصلا اتوك ةينمضلا اتوك ( نيـب اـنراقو ةدـيدجلا مظنلل ةيددعلا لولحلا داجيلإ نيتقيرطلا . ةحيرصلا ةقيرطلل ةيددعلا جئاتنلا نم ةقد رثكأ يه ةينمضلا ةقيرطلل ةيددعلا جئاتنلا نأ انظحلا .


Introduction:
Many physical, chemical and engineering problems, mathematically, can be modeled in the form of system of partial differential equations or system of ordinary differential equations. Parabolic PDEs (partial differential equations) describe practically useful phenomena such as transport-chemistry problems of the advectiondiffusion-reaction type and problem of this type play an important role in the modeling of pollution of the atmosphere, ground water and surface water [9].
Khaddaj and Liddell [6] discussed the numerical solution of time-dependent partial differential equations (PDEs) by the numerical method of lines (NMOL) transporter based systems. Parallel methods for the solution of systems of ordinary differential equations (ODEs) are considered. Algorithms based on the NMOL are parallel ODEs solver are presented.
Celi, R [4] used the method of lines to apply on the equations of helicopter rotor vortex wakes, and converts the governing PDEs into a system of ODEs. These ODEs can then be coupled to other ODEs modeling helicopter dynamics, for time-marching simulations are to extract linearized models. The MOLs is applied to a simplified set of wake equations that has an analytical solution. Because these simplified equations neglect key wake physics, the study is only a first step toward applying MOLs to realistic models. Therefore, the conclusions only apply to the simplified problem considered. He shows that the results of the MOLs are a suitable method to formulate vortex wake models in state-space form. The solutions are accurate and numerically stable. Refining the space discretization increases the stiffness of the ODEs, but explicit solvers can still be used.
Asher et. al. [1] show that the implicit-explicit Range-Kutta schemes are strictly related which provides a framework for the derivation of more general, accurate and efficient schemes.
Minion [10] used method of Lines on PDEs, and then semi-implicit formulation of the method of spectral deferred corrections for ODEs with both stiff and non-stiff terms.
Butcher [3] has developed many types of implicit Rung-Kutta method to reduce storage space and getting more accurate solution.
In this paper we find the numerical solution of system of non-linear PPDEs by transforming the non-linear PPDE to a system of non-linear ordinary differential equation and then using Explicit and implicit Rung-Kutta methods. We use the NMOLs to transform the system of parabolic partial differential equations (PPDEs) into a systems of ODEs, ( [6], [7]), and then we study the properties of the numerical solution of IVPs in ODEs. Also we recall the basic definitions and theories that concern numerical solution of IVPs in ODEs.

The Gray-Scott Model Problem in Two Dimensions
Reaction-diffusion models of chemical species can produce a variety of patterns, reminiscent of those often seen in nature. The Gray-Scott equations model can be consider as reaction. Numerical simulations of this model were performed in an attempt to find stationary lamellar patterns like those observed in earlier laboratory experiments on ferrocyanideiodate-sulphite reactions [13]. The chemical reactions for this situation are described by where U, V and P are chemical species. The system of reaction-diffusion equations for this situation is given by ( ) where u and v are concentrations of two reactions, 1  and 2  are the diffusion rates in the process, g represents the rate of conversion of V to P, and f the rate of the process that feeds U and drains U, V and P ( [14], [12]).
We choose the model parameters as  (1) and (2) From pattern formation the following reaction diffusion system ( [15], [5]) exhibits complicated Solution behavior: where α1=810 -5 and α2=410 -5 The initial conditions are In this model self-replicating spots have been observed. These are regions in which the (chemical) concentrations of some of the species exhibit large amplitude perturbations from a surrounding homogenous state.

Runge-Kutta Methods Solution for ODEs
The general form of s-stage Runge-Kutta method [3] can be written as where s represents the number of stage for the method ci, aij, bi, i,j=,2,…,s are parameters.
The windows form of the method is  [3].
If the coefficients aij are equal to zero for i j  then each kj can be explicitly computed in term of the (i-1) coefficients k1, …, kj-1 and in this case the explicit RK method equation (5)  … (8) and the function kj are defined by a set of simplicity equations then RK method is said to be full IRK method ( [2], [3]).

The Method of Lines:
The method of Lines (MOLs) is a convenient technique for solving timedependent partial differential equations. Replace all spatial derivatives with finite differences, but leave the time derivatives intact, and use an ordinary differential equation solver on the resulting system. In effect, this is an implicit time-stepping finite difference algorithm with the time step determined automatically adaptively by the ODEs solver along a line in time (see figure (1)) [11].
Example: Given a general parabolic equation of the form where L is an elliptic operator. Let Ln be a finite difference operator on a grid xi=a+ih. We can form a semi-discrete system [11] of ordinary differential of the following form In other words, we only discretize the spatial variable. = , the discretize system of ODE [11] is:

Explicit RK method
The form of ERK method of order four is [8] n 1 n 1 2 3 4 h u u (k 2k 2k k ) 6

Implicit Runge-Kutta Method
The general R-stage Implicit RK method is defined by [8]

Numerical Solution of the Model
We find the numerical solution to PPDEs, system (1) and (2)   where xi=ih, yi=jh and i, j, n=1,2,…,N.

Conclusion
The method of lines is convenient technique to transform the system of non-linear parabolic partial differential equations into the systems of ordinary differential equations and thus the new systems require special treatment. When we use ERK method to solve the system of ODEs, we notice that the region of stability is very small but when we used IRK method the stability region is big. Thus from table (1) we conclude that the numerical results of IRK method is more accurate than the results of ERK method, and we found least squares of each method.