American Journal of Computational Mathematics, 2025, 15(3), 246-258 https://www.scirp.org/journal/ajcm ISSN Online: 2161-1211 ISSN Print: 2161-1203 DOI: 10.4236/ajcm.2025.153013 Jul. 30, 2025 246 American Journal of Computational Mathematics Higher Order Approximation of Advection Diffusion Equation by Semi-Discretization Method Khandoker Nasrin Ismet Ara, Md. Sabbir Alam, Laek Sazzad Andallah Department of Mathematics, Jahangirnagar University, Savar, Bangladesh Abstract A system of ordinary differential equations (ODEs) is produced by the semi- discretize method of discretizing the advection diffusion equation (ADE). Runge-Kutta methods of the second and fourth orders are used to solve the system of ODEs. We compute the ADE numerically for initial and boundary conditions, for which the exact solution is known. In the semi-discretization approach, we estimate the error for both the second and fourth-order Runge- Kutta schemes. The semi-discretization method’s outcome is contrasted with the ADE’s numerical solution derived from the complete discretization ex- plicit centered difference scheme. Keywords Advection Diffusion Equation, Semi-Discretization, Finite Difference Scheme, Rate of Convergence, System of ODE’s 1. Introduction The semi-discretization method (SDM) is used to derive numerical solutions for time-dependent partial differential equations (PDEs). The numerical solution procedure is known as space semi-discretization, in which the time variable is left continuous while the space variable is discretised and time integration is applied to transform the PDEs into a system of ordinary differential equations (ODEs). We look at the mathematical model of advection diffusion, which may be used to simulate natural processes in a variety of natural science and engineering applica- tions. Many scholars have tried to solve this equation and apply it in their simu- lations [1]-[8]. A finite difference approach is used to solve the advection diffusion problem. It is well known that the optimal approach to solving the advection dif- How to cite this paper: Ara, K.N.I., Alam, M.S. and Andallah, L.S. (2025) Higher Or- der Approximation of Advection Diffusion Equation by Semi-Discretization Method. American Journal of Computational Math- ematics, 15, 246-258. https://doi.org/10.4236/ajcm.2025.153013 Received: May 19, 2025 Accepted: July 27, 2025 Published: July 30, 2025 Copyright © 2025 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access