Applying Multiquadric Quasi-Interpolation to Solve KdV Equation

Abstract:

      Quasi-interpolation is very useful in the study of approximation theory and its applications, since it can yield solutions directly without the need to solve any linear system of equations. Based on the good performance, Chen and Wu presented a kind of multiquadric (MQ) quasi-interpolation, which is generalized from the $\mathscr{L_D}$ operator, and used it to solve hyperbolic conservation laws and Burgers' equation. In this paper, a numerical scheme is presented based on Chen and Wu's method for solving the Korteweg-de Vries (KdV) equation. The presented scheme is obtained by using the second-order central divided difference of the spatial derivative to approximate the third-order spatial derivative, and the forward divided difference to approximate the temporal derivative, where the spatial derivative is approximated by the derivative of the generalized $\mathscr{L_D}$ quasi-interpolation operator. The algorithm is very simple and easy to implement and the numerical experiments show that it is feasible and valid.