The Group Preserving Scheme For The Forward-Backward Heat Equation

The forward-backward heat conduction equation is widely used in vari-ous fields, for example, fluid dynamics problems of boundary layer, plasma physics, random process theory and the electron beam through solar coronal in astrophysics. Therefore, there are important theoretical value and realis-tic significance in studying numerical methods for the forward-backward heat conduction equation.Because of the important value and the particularity of the initial con-ditions, this equation has attracted a lot of experts and scholars to research it. Houde Han and Dongsheng Yin used the non-overlapping domain decom-position method and the iterative process to solve one-dimensional forward-backward heat conduction equation. They used the forward and backward difference method to calculate in two sub-domains, respectively, and gave the initial value on the inner boundary of two sub-domains. They updated the value of the inner boundary by iterative calculation, finally got an error which controlled the numerical solution. Recently, C.N. Dawson, Qiang Du and T.F. Dupont used domain decomposition and finite difference method to solve the forward-backward heat conduction equation, obtaining the value of the inner boundary by the explicit difference scheme. Once obtaining the value of in-ner boundary, they could use the backward difference method to derive the interior values, finally got higher order error estimate in the maximum norm. Chein-Shan Liu and his co-workers presented forward group preserving scheme (FGPS) and backward group preserving scheme (BGPS) to solve the forward and backward heat conduction equation, and obtained good results. And they also extended these schemes to the wave equation and advection-dispersion equation, respectively. However, they only used one scheme, FGPS or BGP-S, for one type of the heat conduction equation in the whole domain. Also, the coefficient of the time derivative term of the heat conduction equation is constant 1. For the stability analysis of BGPS, they only considered one dimensional case. By using the time transformation, they changed the back-ward heat conduction equation into a forward one, then followed the stability analysis of the forward heat conduction equation to derive the results.This paper is mainly based on the research of Chein-Shan Liu and his co-workers. The differences are as follows:firstly, the coefficient of the time derivative term of the heat conduction equation we considered is σ(x); secondly, we not only can use one scheme, FGPS or BGPS, for one type of the heat conduction equation in the whole domain, but also can simultaneously use FGPS and BGPS in the two sub-domains, respectively; finally, we analyze the stability of both 1D and 2D BGPS, and achieve good results. The way we used is different from that of Liu et al.,This paper consists four chapters.In Chapter 1, we will introduce the backgrounds of the forward-backward heat conduction equation, and the numerical research works of the formers.In Chapter 2, we will introduce dynamics on a future (resp. past) cone and GPS for forward (resp. backward) problems.In Chapter 3, we will establish GPS for 1D forward and backward heat conduction equation. We will analyze the stability and present numerical examples to illustrate our theoretical results.In Chapter 4, Based on the results of Chapter 3, we will extend to provide GPS for 2D forward and backward heat conduction equation. Also, we will analyze the stability and present numerical examples.