The Finite Volume Element Method For Two-Dimensional Burgers Equation

Burgers equation is the simplest nonlinear convection-diffusion model, espe-cially appears in turbulence, heat transfer, mass transfer, air and water pollution and many other areas, while Burgers equation can be used as a simplified mathe-matical model of fluid dynamics equations, and so the subject has broad applicative background as a mathematical model. Therefore, the discussion of the equation has important theoretical and practical significance.The finite volume element method is a discrete technique for the partial dif-ferential equations. Because this method possess the crucial physical conservation properties of the original problem locally, so the finite volume method has been widely used in computational fluid mechanics and heat transfer problems.Since the discontinuous Galerkin method was introduced by Reed and Hill, the approach has been used to solve many partial differential equations. Compare DG method with the finite element method, it's space does not need to meet any continuous conditions, so the structure of the space is simple, and has good locality and parallelism.Based on the advantages of using discontinuous functions as approximation in discontinuous Galerkin methods, it is natural to consider using discontinuous func-tion as trial functions in the finite volume method, which we called the discontinuous finite volume method. Such method has the flexibility of the discontinuous Galerkin method and the simplicity and conservative properties of the finite volume method.In this paper, the following two-dimensional Burgers equation is simulated by two methods called the finite volume method and the discontinuous finite volume method.This paper is organized as follows:In the first chapter, we describe the background and present situation of the research for the Burgers equation at home and abroad, on the other show my work.In the second chapter, we consider the semi-discrete finite volume element scheme and discrete finite volume element scheme for the above two-dimensional Burgers equation, by making the numerical analysis, obtain the optimal error esti-mates of H1-norm about the function.In the third chapter, we consider the semi-discrete discontinuous finite volume element scheme and discrete discontinuous finite volume element scheme for the above two-dimensional Burgers equation, by making the numerical analysis,obtain the optimal error estimates of H1-norm about the function.