Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws: Analysis and application in one dimension | Zendy

Christian Klingenberg | Zendy; Gero Schnücke | Zendy; Yinhua Xia | Zendy

Open Access

Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws: Analysis and application in one dimension

Author(s) -

Christian Klingenberg,

Gero Schnücke,

Yinhua Xia

Publication year - 2015

Publication title -

mathematics of computation

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.95

H-Index - 103

eISSN - 1088-6842

pISSN - 0025-5718

DOI - 10.1090/mcom/3126

Subject(s) - conservation law , mathematics , discontinuous galerkin method , eulerian path , piecewise , mathematical analysis , nonlinear system , scalar (mathematics) , maximum principle , lagrangian , mathematical optimization , geometry , finite element method , optimal control , physics , quantum mechanics , thermodynamics

In this paper, we develop and analyze an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method with a time-dependent approximation space for one dimensional conservation laws, which satisfies the geometric conservation law. For the semi-discrete ALE-DG method, when applied to nonlinear scalar conservation laws, a cell entropy inequality, L2 stability and error estimates are proven. More precisely, we prove the sub-optimal (k + 12) convergence for monotone fluxes, and optimal (k + 1) convergence for an upwind flux, when a piecewise P k polynomial approximation space is used. For the fully-discrete ALE-DG method, the geometric conservation law and the local maximum principle are proven. Moreover we state conditions for slope limiters, which ensure total variation stability of the method. Numerical examples show the capability of the method.

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