Commun. Comput. Phys., 14 (2013), pp. 1174-1206.


Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes

Walter Boscheri 1, Michael Dumbser 1*

1 Laboratory of Applied Mathematics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, I-38123 Trento, Italy.

Received 18 October 2012; Accepted (in revised version) 1 March 2013
Available online 13 June 2013
doi:10.4208/cicp.181012.010313a

Abstract

In this article we present a new class of high order accurate Arbitrary-Eulerian-Lagrangian (ALE) one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes. A WENO reconstruction algorithm is used to achieve high order accuracy in space and a high order one-step time discretization is achieved by using the local space-time Galerkin predictor proposed in [25]. For that purpose, a new element-local weak formulation of the governing PDE is adopted on moving space-time elements. The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes. Moreover, a polynomial mapping defined by the same local space-time basis functions as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element. To maintain algorithmic simplicity, the final ALE one-step finite volume scheme uses moving triangular meshes with straight edges. This is possible in the ALE framework, which allows a local mesh velocity that is different from the local fluid velocity. We present numerical convergence rates for the schemes presented in this paper up to sixth order of accuracy in space and time and show some classical numerical test problems for the two-dimensional Euler equations of compressible gas dynamics.

AMS subject classifications: 65Mxx, 35Lxx

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Key words: Arbitrary Lagrangian-Eulerian, high order reconstruction, WENO, finite volume, local space-time Galerkin predictor, moving unstructured meshes, Euler equations.

*Corresponding author.
Email: walter.boscheri@unitn.it (W. Boscheri), michael.dumbser@unitn.it (M. Dumbser)
 

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