A meshless two-stage scheme for the fifth-order dispersive models in the science of waves on water

Abstract

In this study, we investigate the numerical solution of the higher odd-order PDEs, called the Kawahara and Kawahara-KdV equations, which are main models to study the dynamics of water waves. We combine the two-stage fourth order exponential Rosenbrock integrator with a meshfree scheme based on multiquadric-radial basis function (MQ-RBF) in time and space discretization, respectively. We demonstrate the scheme based on this combination for two test examples of the Kawahara-type dispersive equations with their solitary wave solutions. The obtained results are compared with the previous findings and are figured out graphically. It is shown that the proposed algorithm provides compatible and higher accurate results in order to efficiently illustrate the behavior of solitary waves.