Delta-shaped basis functions-pseudospectral method for numerical investigation of nonlinear generalized equal width equation in shallow water waves

Abstract

In this work, we examine generalized equal width (GEW) equation which is a highly nonlinear partial differential equation and describes plasma waves and shallow water waves. Nonlinearity of the equation is tackled by a linearization technique and finite difference approach is utilized for time derivatives. For spatial derivatives we first introduce delta-shaped basis functions which are relatively less studied in literature. Then, by using delta-shaped basis functions, pseudospectral differentiation matrices are constructed for spatial derivatives. Therefore combining finite difference approach for time derivatives and pseudospectral differentiation matrices for spatial derivatives, we obtain a system of linear equations. Solution of this system of equations gives nodal values of numerical solution of the GEW equation for each time step. Stability of the proposed method is given by using linear matrix stability analysis. To measure performance of the proposed method, four classic test problems are chosen namely the propagation of a single solitary wave, interaction of two solitary waves, Maxwellian initial condition and collision of solitons. Also, conservations of mass, momentum and energy are monitored during simulations. The results of numerical computations are compared with exact results if available and with previous studies in the literature such as Petrov–Galerkin, B-spline Galerkin and some collocation methods. From the comparison we can deduce that the proposed method gives reliable and accurate results in less computational cost.