A meshfree method for the nonlinear KdV equation using stabilized collocation method and gradient reproducing kernel approximations

Abstract

A gradient reproducing kernel based stabilized collocation method (GRKSCM) to numerically solve complicated nonlinear Korteweg-de Vries (KdV) equation is proposed in this paper. The acquisition of GRK through high-order consistency conditions reduces the complexity of RK derivative operations and remarkably improves the effectiveness of the proposed method by directly forming GRK approximations. Owing to the fulfillment of high-order integration constraints, the SCM achieves exact integration in the domain and on boundaries. Von Neumann analysis is utilized to establish the stability criteria for GRKSCM when combined with forward difference temporal discretization. The effectiveness of the GRKSCM method in solving the KdV equation is investigated through six numerical examples. The examples include the motion of the single solitary wave propagation, interaction between two solitary waves and interaction among three solitary waves. Furthermore, the behavior of a two-dimensional solitary wave and the propagation of a three-dimensional solitary wave are also numerically investigated. The numerical outcomes confirm that GRKSCM provides high accuracy in comparison with analytical solutions. In addition, the invariants and error analysis show the conservation of our proposed GRKSCM method.