Discrete singular convolution for the generalized variable-coefficient Korteweg-de Vries equation

Abstract

Numerical solutions of the generalized variable-coefficient Korteweg-de Vries equation are obtained using a discrete singular convolution and a fourth order singly diagonally implicit Runge-Kutta method for space and time discretisation, respectively. The theoretical convergence of the proposed method is rigorously investigated. Test  problems including propagation of single solitons and interaction of solitary waves are performed to verify the efficiency and accuracy of the method. The numerical results are checked against available analytical solutions and compared with the Sinc numerical method. We find that our approach is a very accurate, efficient and reliable method for solving nonlinear partial differential equations.

Key words: Generalized Korteweg-de Vries equations, discrete singular convolution, exponential time differencing methods, soliton solutions.