Six-order compact finite difference method for solving KDV-Burger equation in the application of wave propagations

Iranian Journal of Numerical Analysis and Optimization

مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 27 آذر 1400  

نوع مقاله: Original Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2021.72366.1058

نویسندگان

Kedir Aliyi Koroche ¹؛ Hailu Muleta Chemeda ²

¹Department of Mathematics, College of Natural and Computational Sciences, Ambo University, Ambo, Ethiopia

²Department of Mathematics, Jimma University College of Natural Sciences, Jimma, Ethiopia

چکیده

In this paper, six order compact finite difference method is presented for solving the one-dimensional KdV-Burger equation. First, the given solution domain is discretized using a uniform discretization grid point in a spatial direction. then, using Taylor series expansion we obtain higher-order finite difference discretization of KdV-Burger equation involving with spatial variable and produce a system of non-linear Ordinary differential equation. Then the obtained system of a differential equation is solved by using the fourth-order Runge-Kutta method. To validate the applicability of proposed techniques, four model examples are considered. The stability and convergent analysis of the present method is worked by using Von Neumann stability analysis techniques by supporting the theoretical and mathematical statements in order to verify the accuracy of the present solution. The quality of the attending method has been shown in the sense of root mean square error $L_{2}$ and point-wise maximum absolute error $L_\infty$. This is used to show, how the present method approximates the exact solution very well and how it is quite efficient and practically well suited for solving the KdV-Burger equation. Numerical results of considered examples are presented in terms of $L_2$ and $L_\infty$ in tables. The graph of obtained present numerical and its exact solution are also presented in this paper. The present approximate numeric solvent in the table and graph shows that the numerical solutions are in good agreement with the exact solution for the given model problem. Hence technique is reliable and capable to solve the one-dimensional KdV-Burger equation.