Analysis of Gray Scott’s Model Numerically

In this paper, a two-dimensional nonlinear coupled Gray Scott system is si-mulated with a finite difference scheme and a finite volume technique. Pre and post-processing lead to a new solution called GSmFoam by understand-ing geometry settings and mesh information. The concentration profile changes over time, as does the intensity of the contour patterns. The OpenFoam solver gives you the confidence to compare the pattern result with efficient numerical algorithms on the Gray Scott model.


Introduction
Chemical physics typically uses reaction-diffusion equations to explain temperature and concentration distributions with some pattern formulations [1]. The diffusion term describes the rate of heat and mass transport, while the reaction term describes the rate of heat and mass creation [1] [2]. Nonlinear concepts are frequently referred to as mass action laws because of their broad application [1] [2]. Similar models can also be employed in other applications where transport phenomena are determined by a random movement and where production or consumption conditions should be taken into consideration [1] [2] [3]. A system of reacting chemicals serves as a classic illustration [3] [4]. Population dynamics uses diffusion and reaction terms to characterize the density of a population.
Diffusion describes the random movement of individuals, while reaction describes the reproduction of those individuals. The kind of reactions used may be different depending on the research. For a long time, they were thought of as a result of reacting components or interacting populations, according to Gray of patterns, which is essential for vision. The structure would arise if a chemical system in a stable state were disturbed. These pattern-forming systems are interesting because they remain out of steady-state over lengthy periods of time, which is a feature of biological systems [3] [4] [5] [6]. Systems for analyzing reaction-diffusion measure the impact of chemical reactions distributed over space.
In well-stirred environments, such changes produce non-spatial periodic oscillations. Transport phenomena like diffusion, which spread these oscillations across space, make them significantly more fascinating [6] [7] [8]. Structure development in the activator-inhibitor Gierer-Meinhardt model is thought to need the interaction of these antagonistic feedback processes (the synthesis of new molecules and their dispersion) [8] [9] [10]. Filip Buric [7] investigated the patterns generation in different species at different parameters in term of behavior which emerges population composition and formation of their structure [10] [11] [12]. M. Cronhjart observed replicators possess continuous variability but behave as small molecules emphasis being placed on their interactions which are explained in such that the Gray Scott model and its extensions presented here specify a limited availability of substrate used in replication and therefore introduce competition between species in fuel consumption but unlike completely generalized models elaborated by H. Takagi [11] [12] [13] [14] [15]. Also, H.
Takagi explained spots patterns in a reaction diffusion system with many chemicals. Jeff S McGough invested cubic auto-catalytic reactions while studying Gray

Gray Scott Model Governing Equations
The Gray Scott model is a two dimensional model of diffusion reaction that can produce interesting patterns. Two species i ξ and j ξ interact in the following ways: While j ξ can only be produced by the above reactions involving two values of j ξ and one i ξ , j ξ decays on its own with a rate ( ) Where, •  • The first term , we called as diffusive term which is concentration coefficient (to be constant). Such term shows that first concentration increase in proportion to operator, we say Laplacian (multidimensional, second order derivative explains variation in the gradient). If the quantity of species is higher in neighboring vicinity, the value of i ξ will increase which constitutes diffusion system known as heat equation [9] [13] [19] [23] [24] [28]. • The second term represents nonlinear reaction rate which is . As there is no constant involved in reaction term but strength of the reaction can be adjustified by concentration constants.
• The third term is ( ) •  represents the rate of conversion of j ξ to new. • 0 χ represents the feeds rate.

Numerical Methods
Next, we will go over numerical solutions for the two-dimensional non-linear coupled Gray Scott problem using finite difference approximation. To demonstrate the difference and similarity between the finite difference approximation and the finite volume method (FVM), and to compare the outcomes of the two methods, the coupled system approximation uses finite difference and finite volume. To begin, the domain must be discretized (split) into equal pieces or control volumes (it is not necessary for the length of each segment to be equal, but for simplicity, we use equal segments). Consider a finite domain  [30]. Methods are categorized into two parts: • Implicit Finite Difference coupled with ADI scheme.
• Fully Implicit Finite Volume scheme.

Scheme 01
Scheme procedure can be as follows: Interior Boundary Points: • Such scheme constitutes tridiagonal structure with ( ) • For 0 γ = , scheme can be recognized as fourth order. 1st Point at Left Boundary: For 0 γ = , the linear system in constant c's can be written as: 2nd Point at Right Boundar: For 0 γ = , the linear system in constant c's can be written as:  585  141  459  9  81  3  3  , , , Top Left Point at Boundary: For 0 γ = , the linear system in constant c's can be written as: American Journal of Computational Mathematics  585  141  459  9  81  3  3  ,  ,  ,  ,  ,  ,  512  64  512  32  512  64  512  c  c  c  c  c  c Top Right Point at Boundary: where F and G are nonlinear terms in Equations (11)- (13). Also, the matrices A, B, C and D are (N × N) sparse with triangular nature.

Scheme 02
It is possible to separate the scheme implementation process into two components, as follows: 1) Steady state two dimensional system with source.
2) Transient two dimensional system with source.

Algorithm Strategy for Steady Phenomena • Divide the domain into the finite sized subdomains (finite control volumes)
and each subdomain is represented by a finite number of grid points (like Nodes). • Integrate the governing differential equation (GDE) over each subdomain.

A. A. A. Amin, D. S. Mashat American Journal of Computational Mathematics
• Consider a profile assumption for the dependent variable (like, interpolation function) to evaluate the above integral which expresses the result as an algebraic quantity at the grid points.
Using the assumptions, In a two-dimensional case, the face areas are assumed to be constant and are treated as 1 A = . The distribution of the φ & ϕ in a particular twodimensional scenario is obtained by formulating discretized equations at each grid node of the subdivided domain. To account for boundary conditions, discretized equations must be changed where flux information is available. The boundary-side coefficient is set to zero, and any flux crossing the boundary is added as a new source to any existing u S and p S components. The resulting equations are then solved to obtain the φ & ϕ two-dimensional distribution. American Journal of Computational Mathematics

Algorithm Strategy for Transient Phenomena
To determine the right-hand side of the aforementioned Equation (22) and Equation (23), we must make an assumption regarding the change in P When calculating the time integral, we can utilize the values from the previous step as well as the values from the current step plus the values from the step plus the step after that. The approach can be generalized by using a weighting value between 0 and 1 and writing the integral in the following way: For example, if θ is equal to 0, then scheme is explicit while θ is equal to 1,

Test Problems
We can organize the problem into two parts one dimensional coupled nonlinear system and two dimensional. To demonstrate the effectiveness of the approaches described, we numerically solved the following example.

One Dimensional Coupled Problem
with initial and boundary conditions,

Two Dimensional Coupled Problem
Two dimensional coupled non-linear Gray Scott model system can be written as: A. A. A. Amin, D. S. Mashat

Simulation Setup
The one of the main purposes of this research study is to get good understandings of the simulation of the reaction diffusion model like

Geometry
The geometry consists of the square block with all the boundaries of the square are walls & stationary. Initially, the concentration will be assumed stationary and will be solved on a uniform mesh using the GSmFoam which is designed to solve nonlinear reaction diffusion system oriented problems. The effect of increased mesh resolution and mesh grading at the center of the walls will be investigated. Convert

Mesh
A system in three dimensional cartesian coordinate form can easily solved by the use of OpenFoam solver with all type of geometries as a default parameters

Boundary and Initial Conditions
The case is set up for geometry with mesh generation is completed in which initial fields are designed in 0 folder. Such case is set up to start at time 0 s t = which contains two sub files named as i ξ and j ξ whose initial and boundary

Space Time Discretisation & Solver Selection
We applied finite volume discretisation schemes in the fv (finite volume) Schemes dictionary in the system directory. The specification of the linear equation solvers, tolerances and other algorithm controls is made in the fv (finite volume) Solution dictionary.

Numerical Schemes
The finite volume schemes (fvSchemes) dictionary in the system directory sets the numerical schemes for terms such as derivatives in equations that appear in applications being run. As standard Gaussian finite volume integration is se-

Results
Numerical computations have been performed using the uniform grid. For solving problem 01, we set some parameters which can be seen from data in Tables 1-3 by using implicit schemes.