A new exceptional points method with application to cell-centered Lagrangian schemes and curved meshes

Claisse, Alexandra; Després, Bruno; Labourasse, Emmanuel; Ledoux, Franck

doi:10.1016/j.jcp.2012.02.017

Cited by 18 publications

(22 citation statements)

References 30 publications

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“…The development of Lagrangian numerical methods can start either directly from the conservative quantities such as mass, momentum, and total energy [5,6], or from the nonconservative form of the governing equations, as proposed in [2,4,8]. Furthermore, the location of the physical variables can be different, hence leading either to the cell-centered approach [6,[9][10][11][12], where all variables are located at the cell barycenter, or to the staggered mesh approach [13,14] with the velocity defined at the cell interfaces and the other variables at the cell barycenter.…”

Section: Introductionmentioning

confidence: 99%

An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

Boscheri

2016

Numerical Methods in Fluids

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Summary In this paper, we present a new family of direct arbitrary–Lagrangian–Eulerian (ALE) finite volume schemes for the solution of hyperbolic balance laws on unstructured meshes in multiple space dimensions. The scheme is designed to be high‐order accurate both in space and time, and the mesh motion, which provides the new mesh configuration at the next time step, is taken into account in the final finite volume scheme that is based directly on a space‐time conservation formulation of the governing PDE system. To improve the computational efficiency of the algorithm, high order of accuracy in space is achieved using the a posteriori MOOD limiting strategy that allows the reconstruction procedure to be carried out with only one reconstruction stencil for any order of accuracy. We rely on an element‐local space‐time Galerkin finite element predictor on moving curved meshes to obtain a high‐order accurate one‐step time discretization, while the mesh velocity is computed by means of a suitable nodal solver algorithm that might also be supplemented with a local rezoning procedure to improve the mesh quality. Next, the old mesh configuration at time level tn is connected to the new one at tn + 1 by straight edges, hence providing unstructured space‐time control volumes, on the boundary of which the numerical flux has to be integrated. Here, we adopt a quadrature‐free integration, in which the space‐time boundaries of the control volumes are split into simplex sub‐elements that yield constant space‐time normal vectors and Jacobian matrices. In this way, the integrals over the simplex sub‐elements can be evaluated once and for all analytically during a preprocessing step. We apply the new high‐order direct ALE algorithm to the Euler equations of compressible gas dynamics (also referred to as hydrodynamics equations) as well as to the magnetohydrodynamics equations and we solve a set of classical test problems in two and three space dimensions. Numerical convergence rates are provided up to fifth order of accuracy in 2D and 3D for both hyperbolic systems considered in this paper. Finally, the efficiency of the new method is measured and carefully compared against the original formulation of the algorithm that makes use of a WENO reconstruction technique and Gaussian quadrature formulae for the flux integration: depending on the test problem, the new class of very efficient direct ALE schemes proposed in this paper can run up to ≈12 times faster in the 3D case. Copyright © 2016 John Wiley & Sons, Ltd.

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Section: Introductionmentioning

confidence: 99%

An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

Boscheri

2016

Numerical Methods in Fluids

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“…Furthermore Lagrangian algorithms require a moving mesh framework, with the physical variables either located at the cell barycenter [33,107,90,91,89] or defined at different positions within each control volume [84,85]. The first approach leads to cell-centered Lagrangian schemes, while the latter is usually addressed as the staggered mesh approach, where the velocity is defined at the cell vertices or interfaces and the other variables at the cell center.…”

Section: Introductionmentioning

confidence: 99%

“…A node-centered solver is adopted to evaluate the time derivatives of the fluxes and it can be seen as a multi-dimensional extension of the Generalized Riemann Problem (GRP) methodology used, for example, in the ADER schemes of Titarev and Toro [119,116,117]. Curved meshes have been used in [33], where a cell-centered Lagrangian method which is translation invariant is proposed. Mesh motion usually leads to large element deformation or distortion, hence requiring a mesh quality optimization process during the simulation.…”

Section: Introductionmentioning

confidence: 99%

Direct Arbitrary-Lagrangian–Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws

Boscheri

Loubère

Dumbser

2015

Journal of Computational Physics

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“…For this reason a lot of effort has been made in the last decades concerning the development and improvement of Lagrangian algorithms, see, for example, [4,15,16,65,68,75,87]. Lagrangian schemes are also classified in the literature according to the location of the physical variables on the mesh: in the cell-centered approach [21,[64][65][66]72] all variables are continuously defined inside the cell, whereas in the staggered mesh approach [60,61] the velocity is continuously defined on the dual cell, around the grid nodes, and the other variables on the primary cell.…”

Section: Introductionmentioning

confidence: 99%

“…Cell-centered Lagrangian algorithms have also been presented in [16,23,24] where general multi-dimensional unstructured meshes are used and a node-based finite volume method is adopted to solve a weakly hyperbolic system of conservation laws that is given by the coupling of the evolution equations of the geometry with the equations of the flow field. Claisse et al [21] presented a cell-centered Lagrangian method for curved meshes which is translation invariant, while in a series of papers [62][63][64] Maire proposed first and second order accurate cell-centered Lagrangian schemes in two and three space dimensions on general polygonal grids. The time derivatives of the fluxes are evaluated with a nodecentered solver that can be interpreted as a multi-dimensional extension of the generalized Riemann problem (GRP) methodology used, for example, in the ADER schemes of Titarev and Toro [77,78,80].…”

Section: Introductionmentioning

confidence: 99%

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

Boscheri

Dumbser

2015

J Sci Comput

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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 ). 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 B M. Dumbser 123 J Sci Comput 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 · Quadraturefree flux integration · WENO reconstruction on moving unstructured meshes · High order of accuracy in space and time · Local rezoning · Hydrodynamics

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A new exceptional points method with application to cell-centered Lagrangian schemes and curved meshes

Cited by 18 publications

References 30 publications

An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

Direct Arbitrary-Lagrangian–Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws

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

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