Lagrangian and moving mesh methods for the convection diffusion equation

Chrysafinos, Konstantinos; Walkington, Noel J.

doi:10.1051/m2an:2007053

Cited by 18 publications

(17 citation statements)

References 20 publications

(31 reference statements)

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“…We avoid constants depending on these global geometric quantities, which are typical of the pure Lagrangian approach w = b [10].…”

Section: Discontinuous Galerkin Methods In Time: Exact Integrationmentioning

confidence: 99%

“…We also refer to [20] for the implementation of first-order dG methods in the context of fluid-structure interactions. It is worth comparing our results for dG methods with the pure Lagrangian framework for the advection-dominated diffusion equation on time-independent domains proposed by Chrysafinos and Walkington [10]. Both analyses have some conceptual similarities but the overall purposes are distinct.…”

Section: Introductionmentioning

confidence: 91%

“…This fact has been used for the definition of both (3.5) and (3.6) and it will also be used in the analysis below. Discontinuous maps are proposed in [10] within a Lagrangian approach to reduce the effect of large b for a hyperbolic-type problem on time-independent domains. We are after stability estimates for dG insensitive to the ALE velocity w and the polynomial degree q.…”

Section: Remark 32 (Continuity Of Ale Map) Since the Ale Map Is Timmentioning

confidence: 99%

“…In the extreme case that the domain does not deform, then w = 0 in Ω t = Ω 0 irrespective of the value of b. We refer to [10] for an analysis of the pure Lagrangian approach with emphasis on the convection-dominated diffusion regime. For the study of the ALE approach we consider, as in [1,4,17,16,18,23], a model problem consisting of a prescribed domain deformation Ω t given by an ALE map A t and the scalar advectiondiffusion equation on Ω t with vanishing Dirichlet boundary condition:…”

Section: Introductionmentioning

confidence: 99%

“…In fact, dG schemes able to reproduce (1.6) are unconditionally stable schemes irrespective of the ALE map. In contrast, Chrysafinos and Walkington [10] consider a fixed domain but tackle the notoriously difficult hyperbolic regime µ b L ∞ . In their framework, the ALE velocity w is designed to compensate for large b and is thus chosen to satisfy w ≈ b.…”

Section: Introductionmentioning

confidence: 99%

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Time-Discrete Higher-Order ALE Formulations: Stability

Bonito¹,

Kyza²,

Nochetto³

2013

SIAM J. Numer. Anal.

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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.

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“…We avoid constants depending on these global geometric quantities, which are typical of the pure Lagrangian approach w = b [10].…”

Section: Discontinuous Galerkin Methods In Time: Exact Integrationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 91%

Section: Remark 32 (Continuity Of Ale Map) Since the Ale Map Is Timmentioning

confidence: 99%