Time-discrete higher-order ale formulations: Stability

Andrea Bonito*, Irene Kyza, Ricardo H. Nochetto

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)577-604
Number of pages28
JournalSiam journal on numerical analysis
Issue number1
Publication statusPublished - 2013


  • ALE formulations
  • DG methods in time
  • Discrete Reynold's identities
  • Domain velocity
  • Geometric conservation law
  • Material derivative
  • Moving domains
  • Stability


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