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Staggered Lagrangian Discretization Based on Cell-Centered Riemann Solver and Associated Hydrodynamics Scheme

Published online by Cambridge University Press:  20 August 2015

Pierre-Henri Maire ,
Raphaël Loubère  and
Pavel Váchal
Pierre-Henri Maire*
Affiliation:
CEA-CESTA, BP 2, 33114, Le Barp, France
Raphaël Loubère*
Affiliation:
Institut de Mathématiques de Toulouse (IMT) Universitè Paul-Sabatier and CNRS, Toulouse, France
Pavel Váchal*
Affiliation:
FNSPE, Czech Technical University (CTU) in Prague, Czech Republic
*
Corresponding author.Email:raphael.loubere@math.univ-toulouse.fr
Corresponding author.Email:raphael.loubere@math.univ-toulouse.fr
Email:vachal@galileo.fjfi.cvut.cz
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Abstract

The aim of the present work is to develop a general formalism to derive staggered discretizations for Lagrangian hydrodynamics on two-dimensional unstructured grids. To this end, we make use of the compatible discretization that has been initially introduced by E. J. Caramana et al., in J. Comput. Phys., 146 (1998). Namely, momentum equation is discretized by means of subcell forces and specific internal energy equation is obtained using total energy conservation. The main contribution of this work lies in the fact that the subcell force is derived invoking Galilean invariance and thermodynamic consistency. That is, we deduce a general form of the sub-cell force so that a cell entropy inequality is satisfied. The subcell force writes as a pressure contribution plus a tensorial viscous contribution which is proportional to the difference between the nodal velocity and the cell-centered velocity. This cell-centered velocity is a supplementary degree of freedom that is solved by means of a cell-centered approximate Riemann solver. To satisfy the second law of thermodynamics, the local subcell tensor involved in the viscous part of the subcell force must be symmetric positive definite. This subcell tensor is the cornerstone of the scheme. One particular expression of this tensor is given. A high-order extension of this discretization is provided. Numerical tests are presented in order to assess the efficiency of this approach. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of this scheme.

Keywords

52B1065D1868U0568U07Lagrangian schemeRiemann solverartificial viscosityhydrodynamicscompressible flow
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 4 , October 2011 , pp. 940 - 978
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Addessio, F. L., Carroll, D. E., Dukowicz, J. K., Harlow, F. H., Johnson, J. N., Kashiwa, B. A., Maltrud, M. E. and Ruppel, H. M., CAVEAT: a computer code for fluid dynamics problems with large distortion and internal slip, Technical Report LA-10613-MS, Los Alamos National Laboratory, 1986.Google Scholar
[2]Barth, T. J., Numerical methods for gasdynamic systems on unstructured meshes, in Kroner, D., Ohlberger, M. and Rohde, C., editors, An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, Proceedings of the International School on Theory and Numerics for Conservation Laws, pages 195–284, Berlin, 1997; Lecture Notes in Computational Science and Engineering, Springer.Google Scholar
[3]Barth, T. J. and Jespersen, D. C., The design and application of upwind schemes on unstructured meshes, in AIAA paper 89-0366, 27th Aerospace Sciences Meeting, Reno, Nevada, 1989.Google Scholar
[4]Bauer, A. L., Burton, D. E., Caramana, E. J., Loubère, R., Shashkov, M. J. and Whalen, P. P., The internal consistency, stability, and accuracy of the discrete, compatible formulation of Lagrangian hydrodynamics, J. Comput. Phys., 218(2) (2006), 572–593.CrossRefGoogle Scholar
[5]Ben-Artzi, M. and Falcovitz, J., Generalized Riemann problems in computational fluid dynamics, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK, 2003.Google Scholar
[6]Burbeau-Augoula, A., A node-centered artificial viscosity method for two-dimensional La-grangian hydrodynamics calculations on a staggered grid, Commun. Comput. Phys., 8(4) (2010), 877–900.Google Scholar
[7]Burton, D. E., Exact conservation of energy and momentum in staggered-grid hydrodynamics with arbitrary connectivity, Advances in the Free Lagrange Method, Springer-Verlag, New-York, 1990.Google Scholar
[8]Campbell, J. C. and Shashkov, M. J., A tensor artificial viscosity using a mimetic finite difference algorithm, J. Comput. Phys., 172(4) (2001), 739–765.Google Scholar
[9]Campbell, J. C. and Shashkov, M. J., A compatible Lagrangian hydrodynamics algorithm for unstructured grids, Seluk J. Appl. Math., 4(2) (2003), 53–70.Google Scholar
[10]Caramana, E. J., Burton, D. E., Shashkov, M. J. and Whalen, P. P., The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, J. Comput. Phys., 146(1) (1998), 227–262.Google Scholar
[11]Caramana, E. J. and Shashkov, M. J., Elimination of artificial grid distortion and hourglass-type motions by means of Lagrangian subzonal masses and pressures, J. Comput. Phys., 142(2) (1998), 521–561.Google Scholar
[12]Caramana, E. J., Shashkov, M. J. and Whalen, P. P., Formulations of artificial viscosity for multidimensional shock wave computations, J. Comput. Phys., 144(1) (1998), 70–97.CrossRefGoogle Scholar
[13]Carré, G., Delpino, S., Després, B. and Labourasse, E., A cell-centered Lagrangian hydrodynamics scheme in arbitrary dimension, J. Comput. Phys., 228(14) (2009), 5160–5183.Google Scholar
[14]Christensen, R. B., Godunov methods on a staggered mesh-an improved artificial viscosity, Technical Report UCRL-JC-105269, Lawrence Livermore National Laboratory, 1990.Google Scholar
[15]Després, B. and Mazeran, C., Lagrangian gas dynamics in two dimensions and Lagrangian systems, Arch. Rational Mech. Anal., 178 (2005), 327–372.Google Scholar
[16]Dukowicz, J. K., A general, non-iterative Riemann solver for Godunov’s method, J. Comput. Phys., 61 (1985), 119–137.Google Scholar
[17]Dukowicz, J. K. and Meltz, B., Vorticity errors in multidimensional Lagrangian codes, J. Comput. Phys., 99 (1992), 115–134.Google Scholar
[18]Godunov, S. K., Zabrodine, A., Ivanov, M., Kraiko, A. and Prokopov, G., Rèsolution numèrique des problèmes multidimensionnels de la dynamique des gaz, MIR, 1979.Google Scholar
[19]Kamm, J. R. and Timmes, F. X., On efficient generation of numerically robust Sedov solutions, Technical Report LA-UR-07-2849, Los Alamos National Laboratory, 2007.Google Scholar
[20]Kuropatenko, V. F., On difference methods for the equations of hydrodynamics, in Janenko, N. N., editor, Difference methods for solutions of problems of mathematical physics, Volume I, pages 116–149, Providence, RI, 1967. American Mathematical Society, Translated from the Proceedings of the Steklov Institute of Mathematics, No. 74 (in Russian) by Danskin, J. M..Google Scholar
[21]Loubère, R. and Caramana, E. J., The force/work differencing of exceptional points in the discrete, compatible formulation of Lagrangian hydrodynamics, J. Comput. Phys., 216(1) (2006), 1–18.Google Scholar
[22]Loubère, R. and Shashkov, M. J., A subcell remapping method on staggered polygonal grids for arbitrary-Lagrangian-Eulerian methods, J. Comput. Phys., 209(1) (2005), 105–138.Google Scholar
[23]Luttwak, G. and Falcovitz, J., Slope limiting for vectors: a novel vector limiting algorithm, Conference on Numerical methods for multi-material fluid flows; Pavia University on September 21-25, 2009. Available at www. eucentre.it/multimat09/media/presentazioni_congresso/luttwak.pdf.Google Scholar
[24]Luttwak, G. and Falcovitz, J., Staggered Mesh Godunov (SMG) schemes for ALE hydrodynamics, Numerical methods for multi-material fluid flows; St Catherine’s College Oxford 5th-8th September 2005. Available at www.icfd.reading.ac.uk/talks/Luttwak.ppt.Google Scholar
[25]Maire, P.-H., A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry, J. Comput. Phys., 228(18) (2009), 6882–6915.Google Scholar
[26]Maire, P.-H., A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured mesh, J. Comput. Phys., 228(7) (2009), 2391–2425.Google Scholar
[27]Maire, P.-H., Abgrall, R., Breil, J. and Ovadia, J., A cell-centered Lagrangian scheme for compressible flow problems, SIAM J. Sci. Comput., 29(4) (2007), 1781–1824.Google Scholar
[28]Maire, P.-H. and Nkonga, B., Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics, J. Comput. Phys., 228(3) (2009), 799–821.Google Scholar
[29]Margolin, L., Shashkov, M. and Smolarkiewicz, P., A discrete operator calculus for finite difference approximations, Comput. Meth. Appl. Mech. Eng., 187 (2000), 365–383.Google Scholar
[30]Richtmyer, R. D., Taylor instability in shock acceleration of compressible fluids, Commun. Pure Appl. Math., 13 (1960), 297–319.Google Scholar
[31]Sedov, L. I., Similarity and Dimensional Methods in Mechanics, Academic Press, New York, 1959.Google Scholar
[32]Shashkov, M. J., Conservative Finite-Difference Methods on General Grids, CRC Press, Boca Raton, FL, 1996.Google Scholar
[33]Neumann, J. von and Richtmyer, R. D., A method for the numerical calculations of hydrody-namical shocks, J. Appl. Phys., 21 (1950), 232–238.Google Scholar
[34]Wilkins, M. L., Calculation of elastic plastic flow, in Alder, B., Fernbach, S. and Rotenberg, M., editors, Methods in Computational Physics, volume 3, pp. 211–263, Academic Press, New York, 1964.Google Scholar
[35]Wilkins, M. L., Use of artificial viscosity in multidimensional fluid dynamic calculations, J. Comput. Phys., 36(3) (1980), 381–403.CrossRefGoogle Scholar
[36]Yang, Y, Zhang, Q. and Sharp, D. H., Small amplitude theory of Richtmyer-Meshkov instability, Phys. Fluids, 6(5) (1994), 1856–1873.Google Scholar