The symmetry-preserving difference schemes and exact solutions of some high-dimensional differential equations

Abstract

A direct and effective method is employed to construct the discrete models of high-dimensional generalized Zakharov–Kuznetsov and diffusion–convection equations. Through the compatibility condition, we construct their potential systems, whose local symmetries can be projected into the local and nonlocal symmetries of the original equations. Furthermore, based on the resulting Lie symmetries, the invariant difference models and symmetry-preserving difference models of the original two equations can be derived by using the orthogonal meshes which are uniform in space. Finally, some exact solutions of these two equations are also obtained with their graphic analysis.