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A high-order one-step sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids. (English) Zbl 1433.76137
Summary: We present a one-step high-order cell-centered numerical scheme for solving Lagrangian hydrodynamics equations on unstructured grids. The underlying finite volume discretization is constructed through the use of the sub-cell force concept invoking conservation and thermodynamic consistency. The high-order extension is performed using a one-step discretization, wherein the fluxes are computed by means of a Taylor expansion. The time derivatives of the fluxes are obtained through the use of a node-centered solver which can be viewed as a two-dimensional extension of the Generalized Riemann Problem methodology introduced by Ben-Artzi and Falcovitz.

##### MSC:
 76M99 Basic methods in fluid mechanics
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##### References:
 [1] Barlow, A.J.; Roe, P., A cell centred Lagrangian Godunov scheme for shock hydrodynamics, Comput fluids, 46, 133-136, (2010) · Zbl 1431.76006 [2] Barth TJ. Numerical methods for conservation laws on structured and unstructured meshes. Technical report, VKI lecture series; 2003. [3] Barth TJ, Jespersen DC. The design and application of upwind schemes on unstructured meshes. In AIAA paper 89-0366, 27th aerospace sciences meeting, Reno, Nevada; 1989. [4] Bauer, A.L.; Burton, D.E.; Caramana, E.J.; Loubère, R.; Shashkov, M.J.; Whalen, P.P., The internal consistency, stability, and accuracy of the discrete compatible formulation of Lagrangian hydrodynamics, J comput phys, 218, 572-593, (2006) · Zbl 1161.76538 [5] Ben-Artzi, M.; Falcovitz, J., Generalized Riemann problems in computational fluids dynamics, (2003), Cambridge University Press · Zbl 1017.76001 [6] Botsis J, Deville M. Mécanique des milieux continus. Presses Polytechniques et Universitaires Romandes; 2006. [7] Caramana, E.J.; Burton, D.E.; Shashkov, M.J.; Whalen, P.P., The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, J comput phys, 146, 227-262, (1998) · Zbl 0931.76080 [8] Després, B.; Mazeran, C., Lagrangian gas dynamics in two dimensions and Lagrangian systems, Arch rational mech anal, 178, 327-372, (2005) · Zbl 1096.76046 [9] Donea, J.; Huerta, A.; Ponthot, J-Ph; Rodriguez-Ferran, A., Arbitrary lagrangian – eulerian methods, Encyclopedia of computational mechanics, vol. 1: fundamentals, (2004), John Wiley and Sons, [chapter 14] [10] Dukowicz, J.K.; Meltz, B., Vorticity errors in multidimensional Lagrangian codes, J comput phys, 99, 115-134, (1992) · Zbl 0743.76058 [11] Godlewski, E.; Raviart, P.-A., Hyperbolic systems of conservation laws, (2000), Springer-Verlag · Zbl 1063.65080 [12] Kamm JR, Timmes FX. On efficient generation of numerically robust Sedov solutions. Technical report LA-UR-07-2849. Los Alamos National Laboratory; 2007. [13] Loubère, R.; Shashkov, M.; Wendroff, B., Volume consistency in a staggered Lagrangian hydrodynamics scheme, J comput phys, 227, 3731-3737, (2008) · Zbl 1147.76044 [14] Luttwak G, Falcovitz J. Slope limiting for vectors: a novel vector limiting algorithm. In: Conference on numerical methods for multi-material fluid flows. Pavia University on September 21-25, 2009. . · Zbl 1453.76100 [15] Maire, P.-H., A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes, J comput phys, 228, 7, 2391-2425, (2009) · Zbl 1156.76434 [16] Maire P-H. A unified sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids. Int J Numer Methods Fluids 2010. doi:10.1002/fld.2328. [Published online in Wiley InterScience]. [17] Maire, P.-H.; Abgrall, R.; Breil, J.; Ovadia, J., A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, SIAM J sci comput, 29, 4, 1781-1824, (2007) · Zbl 1251.76028 [18] Maire, P.-H.; Breil, J., A second-order cell-centered Lagrangian scheme for two-dimensional compressible flow problems, Int J numer methods fluids, 56, 8, 1417-1423, (2008) · Zbl 1151.76021 [19] Sod, G.A., A survey of several finite difference methods for systems of non-linear hyperbolic conservation laws, J comput phys, 27, 1-31, (1978) · Zbl 0387.76063 [20] Venkatakrishnan, V., Convergence to steady state solutions of the Euler equations on unstructured grids with limiters, J comput phys, 118, 120-130, (1995) · Zbl 0858.76058
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