Improvement of MacCormack's scheme for Burgers' equation. Using a finite element method

Abstract

SUMMARYA brief description of the finite element method to be used is given. It is then shown how various finite difference schemes for the wave and Burgers' equations can be achieved, in particular the predictorcorrector method of MacCormack. By the finite element method a more efficient predictor-corrector scheme is also obtained. Furthermore, the study tends to show that terms of the general form a(uvw)/dx should be discretized by considering u, v and w as separate variables instead of taking the product uuw as a single variable as is often done with equations written in conservative (divergence law) form.

Semi-discrete and fully discrete HDG methods for Burgers' equation

“…It is widely used in many physical fileds such as fluid mechanics, nonlinear acoustics, and gas dynamics. In recent decades, there have developed many finite element (FE) methods for Burgers' equation, such as conforming methods [1,4,9,10,16,25,29], B-spline methods [3,26,37], least-squares methods [23,35,39], mixed methods [12,19,20,30,33], discontinuous Galerkin (DG) methods [5,32,40], and weak Galerkin methods [14,21].…”

Semi-discrete and fully discrete HDG methods for Burgers' equation

“…In addition, the global spreading of COVID-19 is usually modeled by a system of nonlinear partial differential equations, while the Burgers' equation captures a key challenge of the nonlinear partial differential equations for modeling virus spreading. In recent decades, there have developed many finite element (FE) methods for Burgers' equation, such as conforming methods [1,4,9,10,16,26,30], B-spline methods [3,27,38], least-squares methods [24,36,40], mixed methods [12,20,21,31,34], discontinuous Galerkin (DG) methods [5,33,41], and weak Galerkin methods [14,22].…”