Time-Discrete Higher-Order ALE Formulations: Stability

Abstract

Abstract. Arbitrary Lagrangian Eulerian (ALE) formulations deal with PDEs on deformable domains upon extending the domain velocity from the boundary into the bulk with the purpose of keeping mesh regularity. This arbitrary extension has no effect on the stability of the PDE but may influence that of a discrete scheme. We propose time-discrete discontinuous Galerkin (dG) numerical schemes of any order for a time-dependent advection-diffusion model problem in moving domains, and study their stability properties. The analysis hinges on the validity of the Reynolds' identity for dG. Exploiting the variational structure and assuming exact integration, we prove that our conservative and non-conservative dG schemes are equivalent and unconditionally stable. The same results remain true for piecewise polynomial ALE maps of any degree and suitable quadrature that guarantees the validity of the Reynolds' identity. This approach generalizes the so-called geometric conservation law (GCL) to higher order methods. We also prove that simpler Runge-Kutta-Radau (RKR) methods of any order are conditionally stable, that is subject to a mild ALE constraint on the time steps. Numerical experiments corroborate and complement our theoretical results.

Time-discrete higher order ALE formulations: a priori error analysis

“…This is due to violation of (1.4) which, in turn, leads to a practical scheme with minimal complexity. The aim in this paper is to provide optimal order a priori error estimates for the dG methods proposed in [6]. Our results hinge on the following two main ingredients:…”

“…Since the main obstruction to higher order accuracy is the time discretization, we continue here our analysis of time-discrete discontinuous Galerkin methods (dG) with exact integration and suitable quadrature started in [6]. Exploiting the variational structure of dG, we have been able to prove that the scheme with exact integration is ALE-free stable (stability constants independent of the ALE map) and unconditionally stable, as well as the equivalence between conservative and nonconservative ALE formulations.…”

“…For τ, t ∈ [0, T ] with τ < t, the variational formulation of (1.2) reads At this point we emphasize that we use test functions that do not have vanishing material derivative. We refer to [6] for a discussion of existence, uniqueness and regularity of the solution of (1.1). Despite the important fact that for realistic simulations involving fluids in 3D the ALE method should to be at least second order in time, such methods are very limited in the literature.…”

“…Reynolds' quadrature enforces (1.4), generalizes the concept of GCL, and is responsible for ALE-free stability [6]. We have also been able to examine the practical Runge-Kutta-Radau (RKR) methods, which are dG methods with Radau quadrature, and show their conditional stability for any order, under a mild ALE-dependent restriction on the time steps.…”

On the stability of the ale space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains

“…We can mention [28], [29] and [8]. The last paper is concerned with the stability analysis of the time DGM without space discretization.…”