NUMERICAL SIMULATION OF THE KdV-MKdV EQUATION

In this paper two numerical schemes for the numerical simulation of the nonlinear partial differential equation u_t+6αuu_x+6βu²u_x+u_xxx=0 are implemented by the method of lines (MOL). The first scheme is based on the inverse scattering transform (IST), and the second scheme is a combination of the IST schemes for the Korteweg-de Vries (KdV) and modified KdV (MKdV) equations. The only difference between the two schemes is in the discretization of the nonlinear terms. Numerical experiments have shown that the first scheme is significantly more accurate than the second one. This demonstrates the importance of a proper discretization of nonlinear terms when a numerical method is designed for solving a nonlinear differential equation.