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An Efficient Quadrature-Free Formulation for High Order Arbitrary-Lagrangian–Eulerian ADER-WENO Finite Volume Schemes on Unstructured Meshes

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Abstract

In this paper we present a new and efficient quadrature-free formulation for the family of cell-centered high order accurate direct arbitrary-Lagrangian–Eulerian one-step ADER-WENO finite volume schemes on unstructured triangular and tetrahedral meshes that has been developed by the authors in a recent series of papers (Boscheri et al. in J Comput Phys 267:112–138, 2014; Boscheri and Dumbser in Commun Comput Phys 14:1174–1206, 2013; Boscheri and Dumbser in J Comput Phys 275:484–523, 2014; Dumbser and Boscheri in Comput Fluids 86:405–432, 2013). High order of accuracy in time is obtained by using a local space–time Galerkin predictor on moving curved meshes, while a high order accurate nonlinear WENO method is adopted to produce high order essentially non-oscillatory reconstruction polynomials in space. The mesh is moved at each time step according to the solution of a node solver algorithm that assigns a unique velocity vector to each node of the mesh. A rezoning procedure can also be applied when mesh distortions and deformations become too severe. The space–time mesh is then constructed by straight edges connecting the vertex positions at the old time level \(t^n\) with the new ones at the next time level \(t^{n+1}\), yielding closed space–time control volumes, on the boundary of which the numerical flux must be integrated. This is done here with a new and efficient quadrature-free approach: the space–time boundaries are split into simplex sub-elements, i.e. either triangles in 2D or tetrahedra in 3D. This leads to space–time normal vectors as well as Jacobian matrices that are constant within each sub-element. Within the space–time Galerkin predictor stage that solves the Cauchy problem inside each element in the small, the discrete solution and the flux tensor are approximated using a nodal space–time basis. Since these space–time basis functions are defined on a reference element and do not change, their integrals over the simplex sub-surfaces of the space–time reference control volume can be integrated once and for all analytically during a preprocessing step. The resulting integrals are then used together with the space–time degrees of freedom of the predictor in order to compute the numerical flux that is needed in the finite volume scheme. We apply the high order algorithm presented in this paper to the equations of hydrodynamics obtaining convergence rates up to fourth order of accuracy in space and time. A set of classical Lagrangian test problems has been solved and the results have been compared with the ones given by the original formulation of the algorithm (Boscheri and Dumbser 2013, 2014). The efficiency has been monitored and measured for each test case and the new quadrature-free schemes were up to 3.7 times faster than the ones based on Gaussian quadrature.

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Acknowledgments

The presented research has been financed by the European Research Council (ERC) under the European Union’s Seventh Framework Programme (FP7/2007–2013) with the research project STiMulUs, ERC Grant Agreement No. 278267. The authors acknowledge PRACE for awarding us access to the SuperMUC supercomputer of the Leibniz Rechenzentrum (LRZ) in Munich, Germany.

The authors would like to thank the two anonymous referees for their constructive comments and suggestions, which definetely helped to improve the quality of this manuscript.

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Boscheri, W., Dumbser, M. An Efficient Quadrature-Free Formulation for High Order Arbitrary-Lagrangian–Eulerian ADER-WENO Finite Volume Schemes on Unstructured Meshes. J Sci Comput 66, 240–274 (2016). https://doi.org/10.1007/s10915-015-0019-2

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