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

Abstract

Sixth-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 the Taylor series expansion, we obtain a higher-order finite difference discretization of the KdV-Burger equation involving spatial variables and produce a system of nonlinear ordinary differential equa-tions. 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 L2 and point-wise maximum absolute error L∞. 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 L2 and L∞ 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 of the given model problem. Hence the technique is reliable and capable for solving the one-dimensional KdV-Burger equation.

Keywords

• kdv-equation
• compact finite difference method
• stability analysis
• convergent of method
• root mean square error
• maximum absolute error