## Investigation the Accuracy of Crank Nicolson Methods and their Modified Schemes for One-Dimensional Linear Convection-Reaction-Diffusion Equations

Published: 2024-02-16

Page: 42-56

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

#### How to Cite

Koroche, K. A. (2024). Investigation the Accuracy of Crank Nicolson Methods and their Modified Schemes for One-Dimensional Linear Convection-Reaction-Diffusion Equations. Asian Journal of Mathematics and Computer Research, 31(1), 42–56. https://doi.org/10.56557/ajomcor/2024/v31i18546

### References

Aliyi K, Shiferaw A, Muleta H. radial basis functions based differential quadrature method for one dimensional heat equation. American Journal of Mathematical and Computer Modeling. 2021;6(2):35-42.

DOI: 10.11648/j.ajmcm.20210602.12 Available: https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20210602.12

Wang F, Kehong Z, Imtiaz A, Hijaz A. Gaussian radial basis functions method for linear and nonlinear convection-diffusion models in physical phenomena. Open Physics. 2021;19:69–76. Available:https://doi.org/10.1515/phys-2021-0011

Available:https://www.degruyter.com/document/doi/10.1515/phys- 2021-0011/html?lang=en

Farlow, Stanley J. Partial differential equations for scientists and engineers. Courier Corporation; 1993.

Kaskov SI. Calculation and experimental study of heat exchange in a system of plane-parallel channels with surface intensifiers. Russian Journal of Nonlinear Dynamics. 2021;17(2):211-20. DOI: 10.20537/nd210206, available: http://www.mathnet.ru/rus/agreement

Kuvshinova AN, Tsyganov AV, Tsyganova YV, Garza HT. Parameter identification algorithm for convection-diffusion transport model. In Journal of Physics: Conference Series. 2021;1745(1):012110 DOI:10.1088/1742-6596/1745/1/012110

Matveev MG, Sirota EA, Semenov ME, Kopytin A.V. Verification of the convective diffusion process-based on the analysis of multidimensional time series, In the CEUR Workshop Proceedings. 2017;433-437.

Karahan H. A third-order upwind scheme for the advection-diffusion equation using Spreadsheets. Advance Engineering Software. Elsevier. 2007;38:688–697.

Ding HF, Zhang YX. A new difference scheme with high accuracy and absolute stability for Solving convection-diffusion equations. Journal of Computational and Applied Mathematics. 2009;230(2):600–606.

DOI: 10.1016/j.cam.2008.12.015.

Glushkov EV, Glushkova NV, Chen CS. The semi-analytical solution to heat transfer problems using Fourier transforms technique, radial basis functions, and the method of fundamental solutions. Numerical Heat Transfer, Part B: Fundamentals. 2007;52:409-427.

Fornberg B, Larsson E and Flyer N. Stable computations with gaussian radial basis functions. SIAM Journal of Science and Computation, Elsevier. 2011;33(2):869–892.

Fulger D, Scalas E, Germano G. Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space–space-time-fractional diffusion equation. Phys Rev E. 2008; 77(2); 021122.

Koley U, Risebro NH, Schwab C, Weber F. A multilevel monte carlo finite difference method for random scalar degenerate convection-diffusion equations. Journal of Hyperbolic Differential Equation. World Scientific Publishing Company. 2017;14(3):415–54.

Ahmad H, Khan TA, Stanimirović PS, Chu YM, Ahmad I. Modified variational iteration Algorithm-II: Convergence and applications to diffusion models. Complexity, Hindawi; 2020.

DOI: 10.1155/2020/8841718

Srivastava MH, Ahmad H, Ahmad I, Thounthong P, Khan NM. Numerical simulation of three- dimensional fractional-order convection-diffusion PDEs by a local meshless method. Thermal Science. 2020;210.

Karahan H. Unconditional stable explicit finite difference technique for the advection-diffusion equation using spreadsheets. Advanced Engineering Software. Elsevier. 2007;8:80–6.

Nazir T, Abbas M, Ismail AIM, Majid AA, Rashid A. The numerical solution of advection diffusion problems using a new cubic trigonometric B-splines approach. Applied Mathematical Model, Elsevier. 2016;40:4586–4611.

DOI: 10.1016/j.apm.2015.11.041

Lu L, Zhang Zhiyi. Research on the solution and simulation of the two-dimensional heat equation. 2023 9th International Conference on Virtual Reality (ICVR). IEEE; 2023.

Merga FE, Chemeda HM, Modified crank–nicolson scheme with richardson extrapolation for one-dimensional heat equation. Iranian Journal of Science and Technology, Transactions A: Science, Springer Nature. 2021;1-10.

DOI: 10.1007/s40995-021-01141-0

Koroche KA. Numerical solution for one-dimensional linear types of parabolic partial differential equation and application to heat equation. Mathematics and Computer Science. 2020;5(4):76-85.

DOI: 10.11648/j.mcs.20200504.12

Asrat T, File G, Aga T. Fourth-order stable central difference method for selfadjoint singular perturbation problems, Ethiopian Journal of Science and Technology. 2016;9(1):53-68.

DOI: 10.4314/ejst.v9i1.5

Rashidinia J, Esfahani F, Jamalzadeh S. B-spline collocation approach for solution of klein-gordon equation. International Journal of Mathematical Modeling and Computations. 2013;3(1):25-33. Available:https://ijm2c.ctb.iau.ir/index.php/ijm2c/article/view/article521815.html

Shokofeh S, Rashidinia J. Numerical solution of the hyperbolic telegraph the equation by cubic B- spline collocation method. Applied Mathematics and Computation, Elsevier. 2016;281:28–38.

DOI: 10.1016/j.amc.2016.01.049

Dingeta M, File G, Aga T. Numerical Solution of Second Numerical Solution of Second-Order One Dimensional Linear Hyperbolic Telegraph Equation. Ethiopian Journal of Education and Science. 2018; 4(1).

Mohebbi A, Dehghan M. High-order compact solution of the one-dimensional heat and Advection– diffusion equations. Appl Math Model, Elsevier. 2010;34:3071-3084. DOI:10.1016/j.apm.2010.01.013

Boztosun I, Charafi A, Zerroukat M, Djidjeli K. Thin-plate spline radial basis function scheme for advection-diffusion problems. Electron J Bound Elements. 2002;267–82.

Boztosun, Ismail, Abdullatif Charafi, Dervis Boztosun. Advection-diffusion equation using compactly supported radial basis functions. Meshfree Methods for Partial Differential Equations. 2016;26:63.

Matveev MG, Kopytin AV, Sirota EA. Combined method for identifying the parameters of a distributed dynamic model. In Precise IV International Conferences on Information Technology and Nanotechnology. 2018;1651-1657.

Tsyganov AV, Tsyganova YV, Kuvshinova AN, Tapia GUR. Metaheuristic

algorithms for identification of the convection velocity in the convection-diffusion transport Model. CEU Workshop Proceeding. 2018;188-196.

Waston D. Radial basis function differential quadrature method for the numerical solution of partial differential equations. The Aquila Digital Community, Dissertations; 2017.