The Discontinuous Galerkin Finite Element Method For Two-Dimensional Burgers Equation

As we know, Burgers equation is one of the most simple models of nonlinear convectiondi?usion. However, generally speaking, this well-posed problem can lead to a shock wave, whichmakes it di?cult to solve Burgers equation. Therefore, the research of numerical methods forthis equation is significant in academic meaning and applied value.In 1973, Reed and Hill have introduced discontinuous Galerkin (DG) method firstly. DGmethod is a class of finite element method which uses completely discontinuous basis functions.It is more ?exibility than the classical finite element method. In various adaptive meshes, itcan be simply applied. At present, it is applied in many fields of study. In 1997, Bassi andRebay used DG method to solve the compression Navier–Stokes equation. In 1998, Cockburnand Chi-Wang Shu proposed the local discontinuous Galerkin (LDG) finite element method.In this paper, we study Runge-Kutta discontinuous Galerkin (RKDG) finite elementmethod for solving two-dimensional inviscid Burgers equation with periodic boundary con-dition. Based on consistent mesh, we use the generalized slope limiter and rectangular element.The approximation of space is Q1 element. Moreover, we give stability analysis, error estimates,the implement details as well as numerical experiments and its numerical results. Comparedwith the triangular element, the rectangular element need less degree of freedom. Hence, itis easy to carry out, and the results are also well. Furthermore, we study LDG method forsolving two-dimensional viscid Burgers equation. In theory, we have obtained L2 stability anda priori L~2 error estimates for the semi-discrete LDG method of this equation. Finally, we givenumerical experiments and its numerical results.