Department of Mathematics

University of Houston

Scientific Computing Seminar

Professor Jacques Rappaz
Chaire d'Analyse et Simulation Numérique
Ecole Polytechnique Fédérale de Lausanne
Switzerland

About a convection-diffusion problem
arising in aluminum production

Thursday, February 3, 2011
3:00 PM- 4:00 PM
Room 646 PGH

Abstract: A standard numerical method in order to approach the solution of a time dependent convection-di¤usion equation in $\phi$ transported with velocity ${\bf u}$ , consists to multiply the full equation by a space dependent test function $\psi$ , to integrate it on the computational domain $\Omega$ ; and to discretize it in space with a finite element method and in time with a finite di¤erence scheme. The diffusion term is integrated by part on $\Omega$ , but not the advected term ${\bf u}\cdot{\bf\nabla}\phi$ . In the convection dominated regime, a streamline upwind method SUPG is used in order to stabilize the numerical scheme. In principle, when the flow is incompressible and confined in $\Omega$ , i.e. when div(${\bf u}$ ) = 0 in $\Omega$ and ${\bf u} \cdot {\bf n}$ = 0 on the boundary $\partial \Omega$ , the integral of $\phi$ on the domain $\Omega$ remains constant in time when the source term is vanishing (conservation of the mass balance). However, on a practical point of view, the velocity ${\bf u}$ is often computed with a Navier-Stokes solver which leads to an approximation ${\bf u}_h$ which is not exactly with divergence free. As an unwelcome numerical effect, the mass balance is not conserved when the time goes up. Especially the mass balance defect can be important when the equation is integrated on a long time. In this talk, we propose an original modification of the standard numerical scheme in order to eliminate this defect and we establish some error estimates produced by this scheme.

This seminar is easily accessible to persons with disabilities. For more information or for assistance, please contact the Mathematics Department at 743-3500.