Stability of higher-order ALE-STDGM for nonlinear problems in time-dependent domains

Abstract

In this paper we investigate the stability of the space-time discontinuous Galerkin method for the solution of nonstationary, nonlinear convection-diffusion problem in time-dependent domains. At first we define the continuous problem and reformulate it using the Arbitrary Lagrangian-Eulerian (ALE) method, which replaces the classical partial time derivative by the so called ALE-derivative and an additional convective term. Then the problem is discretized with the aid of the ALE space-time discontinuous Galerkin method (ALE-STDGM). The discretization uses piecewise polynomial functions of degree p ≥ 1 in space and q > 1 in time. Finally in the last part of the paper we present our results concerning the unconditional stability of the method. An important step is the generalization of a discrete characteristic function associated with the approximate solution and the derivation of its properties, namely its continuity in the ∥⋅∥L2 -norm and in special ∥⋅∥ DG -norm.