Commun. Comput. Phys., 14 (2013), pp. 301-327.


On Arbitrary-Lagrangian-Eulerian One-Step WENO Schemes for Stiff Hyperbolic Balance Laws

Michael Dumbser 1*, Ariunaa Uuriintsetseg 1, Olindo Zanotti 1

1 Laboratory of Applied Mathematics, University of Trento, I-38123 Trento, Italy.

Received 29 February 2012; Accepted (in revised version) 24 August 2012
Available online 27 November 2012
doi:10.4208/cicp.310112.120912a

Abstract

In this article we present a new family of high order accurate Arbitrary Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff hyperbolic balance laws. High order accuracy in space is obtained with a standard WENO reconstruction algorithm and high order in time is obtained using the local space-time discontinuous Galerkin method recently proposed in [20]. In the Lagrangian framework considered here, the local space-time DG predictor is based on a weak formulation of the governing PDE on a moving space-time element. For the space-time basis and test functions we use Lagrange interpolation polynomials defined by tensor-product Gauss-Legendre quadrature points. The moving space-time elements are mapped to a reference element using an isoparametric approach, i.e. the space-time mapping is defined by the same basis functions as the weak solution of the PDE. We show some computational examples in one space-dimension for non-stiff and for stiff balance laws, in particular for the Euler equations of compressible gas dynamics, for the resistive relativistic MHD equations, and for the relativistic radiation hydrodynamics equations. Numerical convergence results are presented for the stiff case up to sixth order of accuracy in space and time and for the non-stiff case up to eighth order of accuracy in space and time.

AMS subject classifications: 65Mxx, 35Lxx

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Key words: Arbitrary Lagrangian-Eulerian, finite volume scheme, moving mesh, high order WENO reconstruction, local space-time DG predictor, moving isoparametric space-time elements, stiff relaxation source terms, Euler equations, resistive relativistic MHD equations, relativistic radiation hydrodynamics.

*Corresponding author.
Email: michael.dumbser@ing.unitn.it (M. Dumbser)
 

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