Numerical Approximation to Nonlinear One Dimensional Coupled Reaction Diffusion System (original) (raw)

Numerical study to coupled three dimensional reaction diffusion system

IEEE Access

The numerical solution of reaction diffusion systems may require more computational efforts if the change in concentrations occurs extremely rapid. This is because more time points are needed to resolve the reaction diffusion process accurately. In this article three finite difference implicit schemes are used which are unconditionally stable in order to enhance consistency. Novelty is reported by compact finite difference implicit scheme on reaction diffusion system with higher accuracy measured by L 2 , L∞ & Relativeerror norms. Efficiency is observed by reducing grid space along small temporal steps. CPU performance, transmission capacity along comparison of three schemes shows excellent agreement with analytical solution.

Two-Dimensional Nonlinear Reaction Diffusion Equation with Time Efficient Scheme

American Journal of Computational Mathematics

This research paper represents a numerical approximation to non-linear two-dimensional reaction diffusion equation from population genetics. Since various initial and boundary value problems exist in two-dimensional reaction-diffusion, phenomena are studied numerically by different numerical methods, here we use finite difference schemes to approximate the solution. Accuracy is studied in term of 2 L , L ∞ and relative error norms by random selected grids along time levels for comparison with exact results. The test example demonstrates the accuracy, efficiency and versatility of the proposed schemes. It is shown that the numerical schemes give better solutions. Moreover, the schemes can be easily applied to a wide class of higher dimension nonlinear reaction diffusion equations with a little modification.

Finite Difference Implicit Schemes to Coupled Two-Dimension Reaction Diffusion System

Journal of Applied Mathematics and Physics

In this research article, two finite difference implicit numerical schemes are described to approximate the numerical solution of the two-dimension modified reaction diffusion Fisher's system which exists in coupled form. Finite difference implicit schemes show unconditionally stable and second-order accurate nature of computational algorithm also the validation and comparison of analytical solution, are done through the examples having known analytical solution. It is found that the numerical schemes are in excellent agreement with the analytical solution. We found, second-implicit scheme is much faster than the first with good rate of convergence also we used NVIDA devices to accelerate the computations and efficiency of the algorithm. Numerical results show our proposed schemes with use of HPC (High performance computing) are very efficient and reliable.

Numerical and Analytical Solutions for a Nonlinear Reaction Diffusion System

2016

In this paper, we have used two powerful numerical and analytical methods for solving type of partial differential equations called reaction diffusion system in one dimension. In the numerical method, we use implicit method for discretizing the nonlinear reaction tern and implicit method for linear diffusion term. The Tanh method is used to find the analytical solution for this model. The traveling wave solutions are found for this system using the above methods and for generalized logistic growth with nonlinearity of second order. Comparison of two methods show a good agreement. Keyword: Reaction diffusion system, Tanh method, Finite difference method.

Dynamical Behavior of Nonlinear Coupled Reaction-Diffusion Model: A Numerical Study Utilizing ADI and Staggered Grid Finite Volume Method in Matlab

IEEE Access, 2023

In this research, we present two Numerical algorithms for studying the dynamics of spatially extended coupled nonlinear reaction-diffusion model. The difference in this work is to model a system of reacting and diffusing chemicals, and how to predict the dynamics of ecologically relevant behavior, including chaos. The goal can be achieved by applying a proposed staggered grid finite volume method for solving the Reaction model, rigorously validating the numerical method and grid generation techniques against established results. Additionally, we integrate an Alternating Direction Implicit formulation to compare the efficiency and accuracy of the model's results. Dynamical system analysis, including linear stability analysis, is applied to comprehend the fundamental qualitative features of the inhibitor-activator system inherent in the coupled reaction-diffusion equations. Results, presented in tables and graphical representations, show that the schemes' high accuracy at fourth-order precision in space and second-order in time, with conditional stability. Numerical results showed that the proposed finite volume approach was exceptionally proficient and accurate for tackling the two-dimensional nonlinear coupled reactiondiffusion model. Overall, the dialog highlights the significance of the proposed staggered grid finite approach and calculations for fathoming coupled reaction-diffusion equations are widely studied in biological and chemical systems of nonlinear differential equations.

Comparison of Finite Difference Solution Methods for Reaction Diffusion System in Two Dimensions

AL-Rafidain Journal of Computer Sciences and Mathematics

In this paper, we study three types of finite difference methods, to find the numerical solution of reaction difference systems of PDEs in two dimensions. These methods are ADE, ADI and Hopscotch, where Gray-Scott model in two dimensions has been considered. Our numerical results show that the ADI method produces more accurate and stable solution than ADE method and Hopscotch method is the best because does not involve any tridiagonal matrix. Also we studied the consistency, stability and convergence of the above methods.

Numerical methods for stiff reaction-diffusion systems

Discrete and Continuous Dynamical Systems - Series B, 2007

In a previous study , a class of efficient semi-implicit schemes was developed for stiff reaction-diffusion systems. This method which treats linear diffusion terms exactly and nonlinear reaction terms implicitly has excellent stability properties, and its second-order version, with a name IIF2, is linearly unconditionally stable. In this paper, we present another linearly unconditionally stable method that approximates both diffusions and reactions implicitly using a second order Crank-Nicholson scheme. The nonlinear system resulted from the implicit approximation at each time step is solved using a multi-grid method. We compare this method (CN-MG) with IIF2 for their accuracy and efficiency. Numerical simulations demonstrate that both methods are accurate and robust with convergence using even very large size of time step. IIF2 is found to be more accurate for systems with large diffusion while CN-MG is more efficient when the number of spatial grid points is large.

On non-standard finite difference models of reaction–diffusion equations

Journal of Computational and Applied Mathematics, 2005

Reaction-diffusion equations arise in many fields of science and engineering. Often, their solutions enjoy a number of physical properties. We design, in a systematic way, new non-standard finite difference schemes, which replicate three of these properties. The first property is the stability/instability of the fixed points of the associated space independent equation. This property is preserved by non-standard one-and two-stage theta methods, presented in the general setting of stiff or non-stiff systems of differential equations. Schemes, which preserve the principle of conservation of energy for the corresponding stationary equation (second property) are constructed by non-local approximation of nonlinear reactions. Assembling of theta-methods in the time variable with energy-preserving schemes in the space variable yields non-standard schemes which, under suitable functional relation between step sizes, display the boundedness and positivity of the solution (third property). A spectral method in the space variable coupled with a suitable non-standard scheme in the time variable is also presented. Numerical experiments are provided.

Numerical approximation of a reaction-diffusion system with fast reversible reaction

Chinese Annals of Mathematics, Series B, 2010

We consider the finite volume approximation of a reaction-diffusion system with fast reversible reaction. We deduce from a priori estimates that the approximate solution converges to the weak solution of the reaction -diffusion problem and satisfies estimates which do not depend on the chemical kinetics factor. It follows that the solution converges to the solution of a nonlinear diffusion problem, as the size of the volume elements and the time steps converge to zero while the kinetic rate tends to infinity.