A third (fourth)-order computational scheme for 2D Burgers-type nonlinear parabolic PDEs on a nonuniformly spaced grid network

Abstract

An implicit compact scheme is proposed to approximate the solution of parabolic partial differential equations (PDEs) of Burgers’ type in two dimensions. These nonlinear PDEs are essential because they describe various mechanisms in engineering and physics. The nonlinear convective and advective processes are discretized with high-order accuracy on an arbitrary grid, which results in a family of high-resolution discrete replacements of given PDEs. The essence of the new scheme lies in its compact character and two-level single-cell discretization, so that one discrete equation leads to the accuracy of orders three or four, depending upon the choice of the grid network. The scheme is used for solving celebrated nonlinear PDEs, such as the nondegenerate convection–diffusion equation, the generalized Burgers–Huxley equation, the Buckley–Leverett equation, and the Burgers–Fisher equation. Many computational results are presented to demonstrate the high-resolution character of the newly proposed scheme.