Commun. Comput. Phys., 16 (2014), pp. 718-763.


A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

Raphael Loubere 1*, Michael Dumbser 2, Steven Diot 3

1 CNRS and Institut de Mathematiques de Toulouse (IMT) Universite Paul-Sabatier, Toulouse, France.
2 Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano, 77 - 38123 Trento, Italy.
3 Fluid Dynamics and Solid Mechanics (T-3), Los Alamos National Laboratory, NM 87545 USA.

Received 18 November 2013; Accepted (in revised version) 14 March 2014
Available online 4 July 2014
doi:10.4208/cicp.181113.140314a

Abstract

In this paper, we investigate the coupling of the Multi-dimensional Optimal Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. The Multi-dimensional Optimal Order Detection (MOOD) method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial reconstructions with a posteriori detection and polynomial degree decrementing processes to deal with shock waves and other discontinuities. In the following work, the time discretization is performed with an elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say the optimal high-order of accuracy is reached on smooth solutions, while spurious oscillations near singularities are prevented. The ADER technique permits not only to reduce the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but it also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost, robustness, accuracy and efficiency. The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD is less expensive (memory and/or CPU time), or because it is more accurate for a given grid resolution. A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic partial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements.

AMS subject classifications: 52B10, 65D18, 68U05, 68U07, 65Y05, 65Y20, 65Z05

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Key words: Finite Volume, high-order, conservation law, polynomial reconstruction, ADER, MOOD, hyperbolic PDE, unstructured meshes, finite volume, one-step time discretization, local continuous space-time Galerkin method, WENO, Euler equations, MHD equations, relativistic MHD equations.

*Corresponding author.
Email: raphael.loubere@math.univ-toulouse.fr (R. Loubere), michael.dumbser@unitn.it (M. Dumbser), diot@lanl.gov (S. Diot)
 

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