On the temporal discretizations of convection dominated convection-diusion equations in time-dependent domain

Abstract

This paper presents the numerical analysis of a convection dominated scalar equation with dierent time discretizations in time-dependent domains. The implicit Euler, Crank-Nicolson and backward-dierence methods are used for the temporal discretization. The time-dependent domain is handled by the arbitrary Lagrangian-Eulerian (ALE) approach. In particular, the non-conservative form of the ALE scheme is considered. The Streamline Upwind Petrov-Galerkin (SUPG) nite element method is used for spatial discretization. It is shown that the stability of the fully discrete solution, irrespective of the temporal discretization, is only conditionally stable. The dependence of the numerical solution on the stabilization parameter k is also studied. It is seen that the Crank-Nicolson scheme is less dissipative than the implicit Euler and the backward dierence method. Moreover, the backward dierence scheme is more sensitive to the stabilization parameter k than the other time discretizations.