The Numerical Simulation Of Fluid And Heat Transfer Applying Discontinuous Galerkin Method

Discontinuous Galerkin Method (DGM) was first introduced in 1973 by Reed and Hill in the framework of neutron transport. It has been under rapid development recently and has found its use very quickly in such diverse applications as aeroacoustics, semi-conductor device simulation, turbomachinery, turbulent flows, materials processing, MHD and plasma simulations, and image processing. Since discontinuous Galerkin (DG) methods assume discontinuous approximate solutions, they can be considered as generalizations of finite volume methods. As a consequence, the DG methods incorporate the ideas of numerical fluxes and slope limiters into the finite element framework in a very natural way. The advantages of the DG methods over classical finite difference and finite volume methods are well-documented in the literature: the DG methods work well on arbitrary meshes, result in stable high-order accurate discretization of the convective and diffusive operators, allow for a simple and unambiguous imposition of boundary conditions and are very flexible to parallelization and adaptivity.This paper mainly deals with linear heat conduction and nonlinear heat conduction problems. For the nonlinear problem, a newly developed technique, namely the Homotopy Analysis Method (HAM) is applied to provide an iteration scheme. Many practical problems, such as potential flows, are similar to heat conduction, thus we hope that the studies in this paper can be helpful for solving these problems using DG method.