Investigation the Accuracy of Crank Nicolson Methods and their Modified Schemes for One-Dimensional Linear Convection-Reaction-Diffusion Equations
Kedir Aliyi Koroche *
Department of Mathematics, College of Natural and Computational Sciences, Ambo University, Ambo, Ethiopia.
*Author to whom correspondence should be addressed.
Abstract
In this paper, Crank-Nicolson and Their Modified scheme are applied to find the solution of the convection-reaction-diffusion equation. First, the given solution domain is discretized. Then, using Taylor series expansion, we obtain the difference scheme of the model problem. By rearranging this scheme, we gain proposed techniques. To verify the validity of the proposed techniques, two model illustrations are considered. The stability and convergent analysis of the present scheme is worked by supporting the theoretical and numerical error bound. The accuracy of the present scheme has been shown in the sense of average absolute error, root means square error, and maximum absolute error norms. Then, the accuracy of the present techniques is compared with the accuracy obtained by another method in the literature. The physical behavior of the present results also has been shown in terms of graphs. As we can seen from the results in tables and physical behavior of solution, the present scheme approximates the exact solution veritably well. These scheme is improve the approximation exact solution of convection-reaction-diffusion equation. Hence, the present solution of convection-reaction-diffusion equation is accurate than pre-existing solution.
Keywords: Convection-reaction-diffusion equations, crank nicolson scheme, modifed crank nicolsen scheme, stability, convergent analysis