Numerical solution of complex modified Korteweg–de Vries equation by Petrov–Galerkin method

Abstract

In this paper, a Petrov–Galerkin method is used to derive a numerical scheme for the complex modified Korteweg–de Vries equation (CMKdV), where we have used cubic B-splines as test functions and linear B-splines as trial functions. An implicit midpoint rule is used to discretize in time. A block non-linear pentadiagonal system is obtained. We solve this system by Newton’s method. The resulting scheme has a fourth order accuracy in space direction and second order in time direction. It is unconditionally stable by von Neumann method. The exact soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness of the scheme. The interaction of two solitary waves for different parameters is discussed.