Galerkin scheme for a 2D non-linear PDE's coupled system simulating heat transfer problem

Mohamed Ghattassi¹; Mohamed Boutayeb²; Jean Rodolphe Roche¹

¹ Institut Elie Cartan, University of Lorraine; ² CRAN, University of Lorraine

doi:10.20906/CPS/CILAMCE2015-0627

Resumo

The heat transfer models in participating media for scientific purpose or engineering application such as high temperature processing, thermal insulation., involve coupled radiation and conduction heat equations. The aim of this communication is to present and analyze a numerical method to simulate a combined radiation conduction heat transfer system in a gray, absorbing and non-scattering medium. The model considered is a system of two nonlinear partial differential coupled equations, the first one is a radiation transfer equation (RTE) and the second one a nonlinear parabolic heat conduction equation (CE). Both equations are coupled through a nonlinear source term. The unknown of the RTE equation is the radiation intensity at any time step, position and direction; the unknown of the conduction equation is the temperature at any time step and position. The 3D model was reduced to 2D using symmetry. The poof of existence and uniqueness of a solution of the coupled system is obtained using fixed-point theorem. To compute a numerical solution of the RTE we consider the classical SN discrete ordinate method for the angular approximation and discontinuous Galerkin method in an unstructured mesh for spacial discretization. An approximation of the solution of the CE has been obtained using finite element in the same mesh that the RTE equation and using Crank Nicholson scheme in the time variable. To take account of the non-linearity of the CE we implement a Newton method, we obtain a sequence of semi-linear auxiliary numerical problems. The coupled system is solved using a fixed-point method. We introduce an adequate functional space and norm to prove convergence of the discontinuous Galerkin method used to solve the RTE and we give an error estimate. Under minimal assumptions, convergence of the numerical method for the auxiliary semi-linear heat equation is obtained and an error estimates are given. The convergence proof of the coupled system is a consequence of the application of a discrete fixed-point theor

Palavras-chave: Radiative Transfer Equation; Conduction Equation; High order Dicontinuous Galerkin Method; Finite Element Method