Time-Discrete Higher-Order ALE Formulations: Stability

Type
Article

Authors
Bonito, Andrea
Kyza, Irene
Nochetto, Ricardo H.

KAUST Grant Number
KUS-C1-016-04

Date
2013-01

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 examine this critical issue for higher-order time stepping without space discretization. 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 Reynold's identity for dG. Exploiting the variational structure and assuming exact integration, we prove that our conservative and nonconservative 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 Reynold's identity. This approach generalizes the so-called geometric conservation law to higher-order methods. We also prove that simpler Runge-Kutta-Radau 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. © 2013 Society for Industrial and Applied Mathematics.

Citation
Bonito A, Kyza I, Nochetto RH (2013) Time-Discrete Higher-Order ALE Formulations: Stability. SIAM J Numer Anal 51: 577–604. Available: http://dx.doi.org/10.1137/120862715.

Acknowledgements
This author was partially supported by NSF grant DMS-0914977 and by Award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).This author was partially supported by NSF grants DMS-0807811 and DMS-0807815.This author was partially supported by NSF grants DMS-0807811 and DMS-1109325.

Publisher
Society for Industrial & Applied Mathematics (SIAM)

Journal
SIAM Journal on Numerical Analysis

DOI
10.1137/120862715

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