The Time Discontinuous Galerkin Method For The Forward-backward Heat Equation

The forward-backward heat equation is widely used in various fields [1]. For example:Fluid dynamics problems of boundary layer. Plasma physics, Random process theory and the sun of the electron beam through Solar coronal in Astrophysics. Therefore, the numerical method of forward-backward heat conduction equation has important theoretical value and realistic significance.Because of the important value and the particularity of the initial con-ditions, this equation has attracted a lot of experts and scholars to devote. Hao L. used Galerkin method and weighted Galerkin method to solve the equation[2]. which not only simplified the calculation, but also reduced the regularity of the request. Donald A.F. used Galerkin method[3] and discon-tinuous Galerkin method [5] to solve the this equation, and also got good results. But, he considered only the one-dimensional problem and did not present numerical experiment. Houde Han and Dongsheng Yin used domain decomposition method by iterative way[15] to solve the the equation and so on.We promoted Hao L. and Donald A.F.’s methods in this paper, using the time DG methods and weighted time DG methods to solve the forward-backward heat equations with general coefficient function for the time differ-ential term. And we used the time discontinuous Galerkin method to solve two-dimensional forward-backward heat equation too. Finally, we achieved good results. This paper will research on the following equations:where:ifx∈(-1.0), then σ(x, t)<0; if x∈(0,1), then σ(x, t)>0; and xut-Δu=f(x, y, t),(x, y, t)∈S=(-1,1)×(0,1)×(0,1),This paper consists four chapters:In Chapter1, we will introduce the backgrounds of the forward-backward heat equation, the numerical research works of the formers.In Chapter2, we will introduce the time DG methods of one-dimensional equation, as well as weighted time DG method. Finally, we will give the error analvsis.In Chapter3, on the basis of the analysis for one-dimensional problem, we will promote the time DG method to solve two-dimensional problem. Also we will give the error analysis.In Chapter4. we will provide an a numerical experiment to verify the time DG method valid.