A composite method based on delta‐shaped basis functions and Lie group high‐order geometric integrator for solving Kawahara‐type equations

Abstract

In this paper, we devise a novel method to solve Kawahara‐type equations numerically. In this novel method, for spatial discretization, we use delta‐shaped basis functions and generate differentiation matrices for spatial derivatives of the Kawahara‐type equations. For discretization of temporal variable, we utilize a high‐order geometric numerical integrator based on Lie group methods. For illustration of efficiency of the suggested method, we consider some test problems. We calculate errors and make some comparisons with other results that exist in literature. We also report changes in conservation laws during numerical simulations, and we indicate that the suggested method can preserve the conservation laws pretty good. Outcomes of numerical simulations indicate that the suggested method in this paper is reliable and effective for nonlinear partial differential equations (PDEs).

Novel Numerical Investigations of Some Problems Based on the Darcy–Forchheimer Model and Heat Transfer

“…1. There are several papers concerning the delta shaped sine collocation method (see for example [41][42][43]), but it is well-known that the Chebyshev pseudospectral method gives a much more accurate solution than the Fourier pseudospectral method for problems with non-periodic boundary conditions. This is why we introduced new Chebyshev delta shaped basis functions.…”