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

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• 作者：W. Boscheri ; M. Dumbser
• 关键词：Arbitrary ; Lagrangian–Eulerian (ALE) ; Finite volume schemes ; Quadrature ; free flux integration ; WENO reconstruction on moving unstructured meshes ; High order of accuracy in space and time ; Local rezoning ; Hydrodynamics
• 刊名：Journal of Scientific Computing
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：66
• 期：1
• 页码：240-274
• 全文大小：13,369 KB
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• 作者单位：W. Boscheri (1)
  M. Dumbser (1)
  
  1. Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano 77, 38123, Trento, Italy
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algorithms
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Mathematical and Computational Physics
  
• 出版者：Springer Netherlands
• ISSN：1573-7691

文摘

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. Keywords Arbitrary-Lagrangian–Eulerian (ALE) Finite volume schemes Quadrature-free flux integration WENO reconstruction on moving unstructured meshes High order of accuracy in space and time Local rezoning Hydrodynamics