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A Conservative Lagrangian Scheme for Solving Compressible Fluid Flows with Multiple Internal Energy Equations

Published online by Cambridge University Press:  20 August 2015

Juan Cheng ,
Chi-Wang Shu  and
Qinghong Zeng
Juan Cheng*
Affiliation:
National Key Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
Chi-Wang Shu*
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
Qinghong Zeng*
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
*
Corresponding author.Email address:[email protected]
Email address:[email protected]
Email address:[email protected]
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Abstract

Lagrangian methods are widely used in many fields for multi-material compressible flow simulations such as in astrophysics and inertial confinement fusion (ICF), due to their distinguished advantage in capturing material interfaces automatically. In some of these applications, multiple internal energy equations such as those for electron, ion and radiation are involved. In the past decades, several staggered-grid based Lagrangian schemes have been developed which are designed to solve the internal energy equation directly. These schemes can be easily extended to solve problems with multiple internal energy equations. However such schemes are typically not conservative for the total energy. Recently, significant progress has been made in developing cell-centered Lagrangian schemes which have several good properties such as conservation for all the conserved variables and easiness for remapping. However, these schemes are commonly designed to solve the Euler equations in the form of the total energy, therefore they cannot be directly applied to the solution of either the single internal energy equation or the multiple internal energy equations without significant modifications. Such modifications, if not designed carefully, may lead to the loss of some of the nice properties of the original schemes such as conservation of the total energy. In this paper, we establish an equivalency relationship between the cell-centered discretizations of the Euler equations in the forms of the total energy and of the internal energy. By a carefully designed modification in the implementation, the cell-centered Lagrangian scheme can be used to solve the compressible fluid flow with one or multiple internal energy equations and meanwhile it does not lose its total energy conservation property. An advantage of this approach is that it can be easily applied to many existing large application codes which are based on the framework of solving multiple internal energy equations. Several two dimensional numerical examples for both Euler equations and three-temperature hydrodynamic equations in cylindrical coordinates are presented to demonstrate the performance of the scheme in terms of symmetry preserving, accuracy and non-oscillatory performance.

Keywords

65M0676M20Lagrangian schemeconservativecell-centeredinternal energy equationcompressible fluid flowthree-temperature model
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 5 , November 2012 , pp. 1307 - 1328
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Benson, D. J., Momentum advection on a staggered mesh, J. Comput. Phys., 100 (1992), 143–162.Google Scholar
[2]Breil, J., Galera, S. and Maire, P.-H., Multi-material ALE computation in inertial confinement fusion code CHIC, Comput. Fluids, 46 (2011), 161–167.Google Scholar
[3]Browne, P. L., Integrated gradients: a derivation of some difference forms for the equation of motion for compressible flow in two-dimensional Lagrangian hydrodynamics, using integration of pressures over surfaces, Los Alamos National Laboratory Report LA-2105872-MS, 1986.Google Scholar
[4]Burbeau-Augoula, A., A node-centered artificial viscosity method for two-dimensional Lagrangian hydrodynamics calculations on a staggered grid, Commun. Comput. Phys., 8 (2010), 877–900.CrossRefGoogle Scholar
[5]Campbell, J. C. and Shashkov, M. J., A tensor artificial viscosity using a mimetic finite difference algorithm, J. Comput. Phys., 172 (2001), 739–765.Google Scholar
[6]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 (1998), 227–262.CrossRefGoogle Scholar
[7]Caramana, E. J., Shashkov, M. J. and Whalen, P. P., Formulations of artificial viscosity for multidimensional shock wave computations, J. Comput. Phys., 144 (1998), 70–97.Google Scholar
[8]Carré, G., Delpino, S., Després, B. and Labourasse, E., A cell-centered Lagrangian hydrodynamics scheme in arbitrary dimension, J. Comput. Phys., 228 (2009), 5160–5183.CrossRefGoogle Scholar
[9]Chang, T., Chen, G. and Shen, L., A numerical method for solving the stiff problem in modeling laser target coupling, Comput. Math. Math. Phys., 33 (1993), 805–814.Google Scholar
[10]Cheng, J. and Shu, C.-W., A high order ENO conservative Lagrangian type scheme for the compressible Euler equations, J. Comput. Phys., 227 (2007), 1567–1596.Google Scholar
[11]Cheng, J. and Shu, C.-W., A third order conservative Lagrangian type scheme on curvilinear meshes for the compressible Euler equation, Commun. Comput. Phys., 4 (2008), 1008–1024.Google Scholar
[12]Cheng, J. and Shu, C.-W., A cell-centered Lagrangian scheme with the preservation of symmetry and conservation properties for compressible fluid flows in two-dimensional cylindrical geometry, J. Comput. Phys., 229 (2010), 7191–7206.Google Scholar
[13]Cheng, J. and Shu, C.-W., Improvement on spherical symmetry in two-dimensional cylindrical coordinates for a class of control volume Lagrangian schemes, Commun. Comput. Phys., 11 (2012), 1144–1168.CrossRefGoogle Scholar
[14]Després, B. and Mazeran, C., Lagrangian gas dynamics in two dimensions and Lagrangian systems, Arch. Rational Mech. Anal., 178 (2005), 327–372.Google Scholar
[15]Dukowicz, J. K. and Meltz, B. J. A., Vorticity errors in multidimensional Lagrangian codes, J. Comput. Phys., 99 (1992), 115–134.Google Scholar
[16]Friedman, A., Three-dimensional laser ray tracing on a r-z Lagrangian mesh, Laser Programs Annual Report, UCRL-50021-83 (1983), 3–51.Google Scholar
[17]Harte, J. A., Alley, W. E., Bailey, D. S., Eddleman, J. L. and Zimmerman, G. B., LASNEX-a 2-D physics code for modeling ICF, Lawrence Livermore National Laboratory Report UCRL-LR-105821-96-4 (1996), 150–164.Google Scholar
[18]Kershaw, D. S., Differencing of the diffusion equation in Lagrangian hydrodynamic codes, J. Comput. Phys., 39 (1981), 375–395.Google Scholar
[19]Liu, W., Cheng, J. and Shu, C.-W., High order conservative Lagrangian schemes With Lax-Wendroff type time discretization for the compressible Euler equations, J. Comput. Phys., 228 (2009), 8872–8891.Google Scholar
[20]Maire, P.-H., A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry, J. Comput. Phys., 228 (2009), 6882–6915.Google Scholar
[21]Maire, P.-H., A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes, J. Comput. Phys., 228 (2009), 2391–2425.Google Scholar
[22]Maire, P.-H., A high-order one-step sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids, Comput. Fluids, 46 (2011), 341–347.CrossRefGoogle Scholar
[23]Maire, P.-H., A unified sub-cell force-based discretization for cell-centered Lagrangian hy-drodynamics on polygonal grids, Int. J. Numer. Methods Fluids, 65 (2011), 1281–1294.Google Scholar
[24]Maire, P.-H., Abgrall, R., Breil, J. and Ovadia, J., A cell-centered Lagrangian scheme for compressible flow problems, SIAM J. Sci. Comput., 29 (2007), 1781–1824.CrossRefGoogle Scholar
[25]Maire, P.-H., Loubère, R. and Vachal, P., Staggered Lagrangian discretization based on cell-centered Riemann solver and associated hydrodynamics scheme, Commun. Comput. Phys., 10(2011), 940–978.Google Scholar
[26]Maire, P.-H. and Nkonga, B., Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics, J. Comput. Phys., 228 (2009), 799–821.Google Scholar
[27]Marinak, M. M., Haan, S. W. and Tipton, R. E., Three-dimensional simulations of national ignition facility capsule implosions with HYDRA, Lawrence Livermore National Laboratory Report UCRL-LR-105821-96-4 (1996), 143–149.Google Scholar
[28]Munz, C. D., On Godunov-type schemes for Lagrangian gas dynamics, SIAM J. Numer. Anal., 31 (1994), 17–42.Google Scholar
[29]von Neumann, J. and Richtmyer, R. D., A method for the calculation of hydrodynamics shocks, J. Appl. Phys., 21 (1950), 232–237.Google Scholar
[30]Pei, W. B., The construction of simulation algorithms for laser fusion, Commun. Comput. Phys., 2 (2007), 255–270.Google Scholar
[31]Sedov, L. I., Similarity and Dimensional Methods in Mechanics, Academic Press, New York, 1959.Google Scholar
[32]Wang, G. and Chang, T., Laser X-ray conversion and electron thermal conductivity, Plasma Sci. Tech., 3 (2001), 653–658.Google Scholar