An efficient numerical method for solving the Korteveg–de Vries equation in a class of discontinuous functions

摘要

In this study, an efficient numerical method is proposed and implemented for solving the well-known one-dimensional Korteveg–de Vries (KdV) equation in a class of discontinuous functions. To this end, an auxiliary problem is formulated and it is shown that the solution of the auxiliary problem is the weak solution of KdV equation. Furthermore, some properties of the solution to the auxiliary problem are established. Since no smoothness assumptions are imposed on the unknown function, the numerical method proposed in this paper gives better results as compared to the conventional finite difference methods. Indeed, several simulations are presented in which the analytical solution and the computed numerical solution are in perfect agreement. Further numerical experiments with initial conditions of varying regularities are provided.