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An efficient quadrature-free formulation for high order arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes on unstructured meshes. (English) Zbl 1338.65219
Summary: 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 (cf. [W. Boscheri et al., “Lagrangian ADER-WENO finite volume schemes on unstructured triangular meshes based on genuinely multidimensional HLL Riemann solvers ”, arXiv:1312.0436], [W. Boscheri and M. Dumbser, “Arbitrary-Lagrangian-Eulerian one-step WENO finite volume schemes on unstructured triangular meshes ”, arXiv:1302.3076; “A direct arbitrary Lagrangian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and nonconservative hyperbolic systems in 3D”, J. Comput. Phys. 275,484–523 (2014)], [M. Dumbser and W. Boscheri, “High-order unstructured Lagrangian one-step WENO finite volume schemes for non-conservative hyperbolic systems: applications to compressible multi-phase flows ”, arXiv:1304.4816]). 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 (cf. [Boscheri and Dumbser, loc. cit.]). 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.

MSC:
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76N15 Gas dynamics (general theory)
76M12 Finite volume methods applied to problems in fluid mechanics
Software:
ReALE; RIEMANN; HE-E1GODF
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References:
[1] Balsara, DS, Multidimensional HLLE Riemann solver: application to Euler and magnetohydrodynamic flows, J. Comput. Phys., 229, 1970-1993, (2010) · Zbl 1303.76140
[2] Balsara, DS, A two-dimensional HLLC Riemann solver for conservation laws: application to Euler and magnetohydrodynamic flows, J. Comput. Phys., 231, 7476-7503, (2012) · Zbl 1284.76261
[3] Balsara, DS, Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics, J. Comput. Phys., 231, 7504-7517, (2012)
[4] Benson, DJ, Computational methods in Lagrangian and Eulerian hydrocodes, Comput. Methods Appl. Mech. Eng., 99, 235-394, (1992) · Zbl 0763.73052
[5] Berndt, M; Breil, J; Galera, S; Kucharik, M; Maire, PH; Shashkov, M, Two-step hybrid conservative remapping for multimaterial arbitrary Lagrangian-Eulerian methods, J. Comput. Phys., 230, 6664-6687, (2011) · Zbl 1408.65077
[6] Berndt, M; Kucharik, M; Shashkov, MJ, Using the feasible set method for rezoning in ALE, Procedia Comput. Sci., 1, 1879-1886, (2010)
[7] Bochev, P; Ridzal, D; Shashkov, MJ, Fast optimization-based conservative remap of scalar fields through aggregate mass transfer, J. Comput. Phys., 246, 37-57, (2013) · Zbl 1349.65054
[8] Boscheri, W; Balsara, DS; Dumbser, M, Lagrangian ADER-WENO finite volume schemes on unstructured triangular meshes based on genuinely multidimensional HLL Riemann solvers, J. Comput. Phys., 267, 112-138, (2014) · Zbl 1349.76309
[9] Boscheri, W; Dumbser, M, Arbitrary-Lagrangian-Eulerian one-step WENO finite volume schemes on unstructured triangular meshes, Commun. Comput. Phys., 14, 1174-1206, (2013) · Zbl 1388.65075
[10] Boscheri, W; Dumbser, M, A direct arbitrary-Lagrangian-Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and nonconservative hyperbolic systems in 3D, J. Comput. Phys., 275, 484-523, (2014) · Zbl 1349.76310
[11] Boscheri, W; Dumbser, M; Balsara, DS, High order Lagrangian ADER-WENO schemes on unstructured meshes—application of several node solvers to hydrodynamics and magnetohydrodynamics, Int. J. Numer. Methods Fluids, 76, 737-778, (2014)
[12] Boscheri, W; Dumbser, M; Zanotti, O, High order cell-centered Lagrangian-type finite volume schemes with time-accurate local time stepping on unstructured triangular meshes, J. Comput. Phys., 291, 120-150, (2015) · Zbl 1349.76311
[13] Boscheri, W., Loubère, R., Dumbser, M.: Direct Arbitrary-Lagrangian-Eulerian ADER-MOOD Finite Volume Schemes for Multidimensional Hyperbolic Conservation Laws. J. Comput. Phys. (2015). doi:10.1016/j.jcp.2015.03.015 · Zbl 1349.76312
[14] Breil, J; Harribey, T; Maire, PH; Shashkov, MJ, A multi-material reale method with MOF interface reconstruction, Comput. Fluids, 83, 115-125, (2013) · Zbl 1290.76094
[15] Caramana, EJ; Burton, DE; Shashkov, MJ; Whalen, PP, The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, J. Comput. Phys., 146, 227-262, (1998) · Zbl 0931.76080
[16] Carré, G; Pino, S; Després, B; Labourasse, E, A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension, J. Comput. Phys., 228, 5160-5183, (2009) · Zbl 1168.76029
[17] Cesenek, J; Feistauer, M; Horacek, J; Kucera, V; Prokopova, J, Simulation of compressible viscous flow in time-dependent domains, Appl. Math. Comput., 219, 7139-7150, (2013) · Zbl 1426.76233
[18] Cheng, J; Shu, CW, A high order ENO conservative Lagrangian type scheme for the compressible Euler equations, J. Comput. Phys., 227, 1567-1596, (2007) · Zbl 1126.76035
[19] Cheng, J; Toro, EF, A 1D conservative Lagrangian ADER scheme, Chin. J. Comput. Phys., 30, 501-508, (2013)
[20] Clain, S; Diot, S; Loubère, R, A high-order finite volume method for systems of conservation laws—multi-dimensional optimal order detection (MOOD), J. Comput. Phys., 230, 4028-4050, (2011) · Zbl 1218.65091
[21] Claisse, A; Després, B; Labourasse, E; Ledoux, F, A new exceptional points method with application to cell-centered Lagrangian schemes and curved meshes, J. Comput. Phys., 231, 4324-4354, (2012) · Zbl 1426.76350
[22] Cockburn, B., Karniadakis, G.E., Shu, C.W.: Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2000)
[23] Després, B; Mazeran, C, Symmetrization of Lagrangian gas dynamic in dimension two and multimdimensional solvers, C. R. Mecanique, 331, 475-480, (2003) · Zbl 1293.76089
[24] Després, B; Mazeran, C, Lagrangian gas dynamics in two-dimensions and Lagrangian systems, Arch. Ration. Mech. Anal., 178, 327-372, (2005) · Zbl 1096.76046
[25] Diot, S; Clain, S; Loubère, R, Improved detection criteria for the multi-dimensional optimal order detection (mood) on unstructured meshes with very high-order polynomials, Comput. Fluids, 64, 43-63, (2012) · Zbl 1365.76149
[26] Diot, S; Loubère, R; Clain, S, The MOOD method in the three-dimensional case: very-high-order finite volume method for hyperbolic systems, Int. J. Numer. Methods Fluids, 73, 362-392, (2013) · Zbl 1218.65091
[27] Dobrev, VA; Ellis, TE; Kolev, TV; Rieben, RN, Curvilinear finite elements for Lagrangian hydrodynamics, Int. J. Numer. Methods Fluids, 65, 1295-1310, (2011) · Zbl 1255.76075
[28] Dobrev, VA; Ellis, TE; Kolev, TV; Rieben, RN, High order curvilinear finite elements for Lagrangian hydrodynamics, SIAM J. Sci. Comput., 34, 606-641, (2012) · Zbl 1255.76076
[29] Dobrev, VA; Ellis, TE; Kolev, TV; Rieben, RN, High order curvilinear finite elements for axisymmetric Lagrangian hydrodynamics, Comput. Fluids, 83, 58-69, (2013) · Zbl 1290.76061
[30] Balsara, DS; Dumbser, M; Abgrall, R, Multidimensional HLLC Riemann solver for unstructured meshes—with application to Euler and MHD flows, J. Comput. Phys., 261, 172-208, (2014) · Zbl 1349.76426
[31] Dubcova, L; Feistauer, M; Horacek, J; Svacek, P, Numerical simulation of interaction between turbulent flow and a vibrating airfoil, Comput. Vis. Sci., 12, 207-225, (2009) · Zbl 1426.74127
[32] Dubiner, M, Spectral methods on triangles and other domains, J. Sci. Comput., 6, 345-390, (1991) · Zbl 0742.76059
[33] Dumbser, M, Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes with time-accurate local time stepping for hyperbolic conservation laws, Comput. Methods Appl. Mech. Eng., 280, 57-83, (2014) · Zbl 1423.76296
[34] Dumbser, M; Balsara, DS; Toro, EF; Munz, C-D, A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes, J. Comput. Phys., 227, 8209-8253, (2008) · Zbl 1147.65075
[35] Dumbser, M; Boscheri, W, High-order unstructured Lagrangian one-step WENO finite volume schemes for non-conservative hyperbolic systems: applications to compressible multi-phase flows, Comput. Fluids, 86, 405-432, (2013) · Zbl 1290.76081
[36] Dumbser, M; Enaux, C; Toro, EF, Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, J. Comput. Phys., 227, 3971-4001, (2008) · Zbl 1142.65070
[37] Dumbser, M; Hidalgo, A; Castro, M; Parés, C; Toro, EF, FORCE schemes on unstructured meshes II: non-conservative hyperbolic systems, Comput. Methods Appl. Mech. Eng., 199, 625-647, (2010) · Zbl 1227.76043
[38] Dumbser, M; Käser, M, Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J. Comput. Phys., 221, 693-723, (2007) · Zbl 1110.65077
[39] Dumbser, M; Käser, M; Titarev, VA; Toro, EF, Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, J. Comput. Phys., 226, 204-243, (2007) · Zbl 1124.65074
[40] Dumbser, M; Toro, EF, On universal osher-type schemes for general nonlinear hyperbolic conservation laws, Commun. Comput. Phys., 10, 635-671, (2011) · Zbl 1373.76125
[41] Dumbser, M; Uuriintsetseg, A; Zanotti, O, On arbitrary-Lagrangian-Eulerian one-step WENO schemes for stiff hyperbolic balance laws, Commun. Comput. Phys., 14, 301-327, (2013) · Zbl 1373.76126
[42] Dumbser, M; Zanotti, O; Loubère, R; Diot, S, A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws, J. Comput. Phys., 278, 47-75, (2014) · Zbl 1349.65448
[43] Feistauer, M; Horacek, J; Ruzicka, M; Svacek, P, Numerical analysis of flow-induced nonlinear vibrations of an airfoil with three degrees of freedom, Comput. Fluids, 49, 110-127, (2011) · Zbl 1271.76165
[44] Feistauer, M., Kucera, V., Prokopova, J., Horacek, J.: The ALE discontinuous Galerkin method for the simulation of air flow through pulsating human vocal folds. In: AIP Conference Proceedings, vol. 1281, pp. 83-86 (2010) · Zbl 1365.76129
[45] Francois, MM; Shashkov, MJ; Masser, TO; Dendy, ED, A comparative study of multimaterial Lagrangian and Eulerian methods with pressure relaxation, Comput. Fluids, 83, 126-136, (2013) · Zbl 1290.76133
[46] Galera, S; Maire, PH; Breil, J, A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction, J. Comput. Phys., 229, 5755-5787, (2010) · Zbl 1346.76105
[47] Hidalgo, A; Dumbser, M, ADER schemes for nonlinear systems of stiff advection-diffusion-reaction equations, J. Sci. Comput., 48, 173-189, (2011) · Zbl 1221.65231
[48] Hirt, C; Amsden, A; Cook, J, An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J. Comput. Phys., 14, 227253, (1974) · Zbl 0292.76018
[49] Hu, C; Shu, CW, Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150, 97-127, (1999) · Zbl 0926.65090
[50] Kamm, J.R., Timmes, F.X.: On efficient generation of numerically robust Sedov solutions. Technical Report LA-UR-07-2849, LANL (2007) · Zbl 1142.65070
[51] Karniadakis, G.E., Sherwin, S.J.: Spectral/hp Element Methods in CFD. Oxford University Press, Oxford (1999) · Zbl 0954.76001
[52] Kidder, RE, Laser-driven compression of hollow shells: power requirements and stability limitations, Nucl. Fusion, 1, 3-14, (1976)
[53] Knupp, PM, Achieving finite element mesh quality via optimization of the jacobian matrix norm and associated quantities. part II—a framework for volume mesh optimization and the condition number of the Jacobian matrix, Int. J. Numer. Methods Eng., 48, 1165-1185, (2000) · Zbl 0990.74069
[54] Kucharik, M; Breil, J; Galera, S; Maire, PH; Berndt, M; Shashkov, MJ, Hybrid remap for multi-material ALE, Comput. Fluids, 46, 293-297, (2011) · Zbl 1433.76133
[55] Kucharik, M; Shashkov, MJ, One-step hybrid remapping algorithm for multi-material arbitrary Lagrangian-Eulerian methods, J. Comput. Phys., 231, 2851-2864, (2012) · Zbl 1323.74108
[56] Li, Z; Yu, X; Jia, Z, The cell-centered discontinuous Galerkin method for Lagrangian compressible Euler equations in two dimensions, Comput. Fluids, 96, 152-164, (2014) · Zbl 1391.76347
[57] Liska, R; Shashkov, MJ; Váchal, P; Wendroff, B, Synchronized flux corrected remapping for ALE methods, Comput. Fluids, 46, 312-317, (2011) · Zbl 1433.76135
[58] Liu, W; Cheng, J; Shu, CW, High order conservative Lagrangian schemes with Lax-Wendroff type time discretization for the compressible Euler equations, J. Comput. Phys., 228, 8872-8891, (2009) · Zbl 1287.76181
[59] Loubère, R; Dumbser, M; Diot, S, A new family of high order unstructured mood and ader finite volume schemes for multidimensional systems of hyperbolic conservation laws, Commun. Comput. Phys., 16, 718-763, (2014) · Zbl 1373.76137
[60] Loubère, R; Maire, PH; Váchal, P, A second-order compatible staggered Lagrangian hydrodynamics scheme using a cell-centered multidimensional approximate Riemann solver, Procedia Comput. Sci., 1, 1931-1939, (2010)
[61] Loubère, R; Maire, PH; Váchal, P, 3D staggered Lagrangian hydrodynamics scheme with cell-centered Riemann solver-based artificial viscosity, Int. J. Numer. Methods Fluids, 72, 22-42, (2013)
[62] Maire, PH, A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes, J. Comput. Phys., 228, 2391-2425, (2009) · Zbl 1156.76434
[63] Maire, PH, A high-order one-step sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids, Comput. Fluids, 46, 341-347, (2011) · Zbl 1433.76137
[64] Maire, PH, A unified sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids, Int. J. Numer. Methods Fluids, 65, 1281-1294, (2011) · Zbl 1429.76089
[65] Maire, PH; Abgrall, R; Breil, J; Ovadia, J, A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, SIAM J. Sci. Comput., 29, 1781-1824, (2007) · Zbl 1251.76028
[66] Maire, PH; Breil, J, A second-order cell-centered Lagrangian scheme for two-dimensional compressible flow problems, Int. J. Numer. Methods Fluids, 56, 1417-1423, (2007) · Zbl 1151.76021
[67] Maire, PH; Nkonga, B, Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics, J. Comput. Phys., 228, 799-821, (2009) · Zbl 1156.76039
[68] Munz, CD, On Godunov-type schemes for Lagrangian gas dynamics, SIAM J. Numer. Anal., 31, 17-42, (1994) · Zbl 0796.76057
[69] Noh, WF, Errors for calculations of strong shocks using artificial viscosity and an artificial heat flux, J. Comput. Phys., 72, 78-120, (1987) · Zbl 0619.76091
[70] Ortega, AL; Scovazzi, G, A geometrically-conservative, synchronized, flux-corrected remap for arbitrary Lagrangian-Eulerian computations with nodal finite elements, J. Comput. Phys., 230, 6709-6741, (2011) · Zbl 1284.76255
[71] Peery, JS; Carroll, DE, Multi-material ALE methods in unstructured grids, Comput. Methods Appl. Mech. Eng., 187, 591-619, (2000) · Zbl 0980.74068
[72] Sambasivan, SK; Shashkov, MJ; Burton, DE, A finite volume cell-centered Lagrangian hydrodynamics approach for solids in general unstructured grids, Int. J. Numer. Methods Fluids, 72, 770-810, (2013)
[73] Sambasivan, SK; Shashkov, MJ; Burton, DE, Exploration of new limiter schemes for stress tensors in Lagrangian and ALE hydrocodes, Comput. Fluids, 83, 98-114, (2013) · Zbl 1290.76107
[74] Scovazzi, G, Lagrangian shock hydrodynamics on tetrahedral meshes: a stable and accurate variational multiscale approach, J. Comput. Phys., 231, 8029-8069, (2012)
[75] Smith, RW, AUSM(ALE): a geometrically conservative arbitrary Lagrangian-Eulerian flux splitting scheme, J. Comput. Phys., 150, 268286, (1999) · Zbl 0936.76046
[76] Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall Inc., Englewood Cliffs (1971) · Zbl 0379.65013
[77] Titarev, VA; Toro, EF, ADER: arbitrary high order Godunov approach, J. Sci. Comput., 17, 609-618, (2002) · Zbl 1024.76028
[78] Titarev, VA; Toro, EF, ADER schemes for three-dimensional nonlinear hyperbolic systems, J. Comput. Phys., 204, 715-736, (2005) · Zbl 1060.65641
[79] Titarev, VA; Tsoutsanis, P; Drikakis, D, WENO schemes for mixed-element unstructured meshes, Commun. Comput. Phys., 8, 585-609, (2010) · Zbl 1364.76121
[80] Toro, EF; Titarev, VA, Derivative Riemann solvers for systems of conservation laws and ADER methods, J. Comput. Phys., 212, 150-165, (2006) · Zbl 1087.65590
[81] Toro, EF, Anomalies of conservative methods: analysis, numerical evidence and possible cures, Int. J. Comput. Fluid Dyn., 11, 128-143, (2002)
[82] Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, Berlin (2009) · Zbl 1227.76006
[83] Tsoutsanis, P; Titarev, VA; Drikakis, D, WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions, J. Comput. Phys., 230, 1585-1601, (2011) · Zbl 1210.65160
[84] Vilar, F, Cell-centered discontinuous Galerkin discretization for two-dimensional Lagrangian hydrodynamics, Comput. Fluids, 64, 64-73, (2012) · Zbl 1365.76129
[85] Vilar, F; Maire, PH; Abgrall, R, Cell-centered discontinuous Galerkin discretizations for two-dimensional scalar conservation laws on unstructured grids and for one-dimensional Lagrangian hydrodynamics, Comput. Fluids, 46, 498-604, (2010) · Zbl 1433.76093
[86] Vilar, F; Maire, PH; Abgrall, R, A discontinuous Galerkin discretization for solving the two-dimensional gas dynamics equations written under total Lagrangian formulation on general unstructured grids, J. Comput. Phys., 276, 188-234, (2014) · Zbl 1349.76278
[87] Neumann, J; Richtmyer, RD, A method for the calculation of hydrodynamics shocks, J. Appl. Phys., 21, 232-237, (1950) · Zbl 0037.12002
[88] Yanilkin, YV; Goncharov, EA; Kolobyanin, VY; Sadchikov, VV; Kamm, JR; Shashkov, MJ; Rider, WJ, Multi-material pressure relaxation methods for Lagrangian hydrodynamics, Comput. Fluids, 83, 137-143, (2013) · Zbl 1290.76138
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