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Improvement on Spherical Symmetry in Two-Dimensional Cylindrical Coordinates for a Class of Control Volume Lagrangian Schemes

Published online by Cambridge University Press:  20 August 2015

Juan Cheng  and
Chi-Wang Shu
Juan Cheng*
Affiliation:
Laboratory of 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
*
Email:[email protected]
Corresponding author.Email:[email protected]
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Abstract

In, Maire developed a class of cell-centered Lagrangian schemes for solving Euler equations of compressible gas dynamics in cylindrical coordinates. These schemes use a node-based discretization of the numerical fluxes. The control volume version has several distinguished properties, including the conservation of mass, momentum and total energy and compatibility with the geometric conservation law (GCL). However it also has a limitation in that it cannot preserve spherical symmetry for one-dimensional spherical flow. An alternative is also given to use the first order area-weighted approach which can ensure spherical symmetry, at the price of sacrificing conservation of momentum. In this paper, we apply the methodology proposed in our recent work to the first order control volume scheme of Maire in to obtain the spherical symmetry property. The modified scheme can preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid, and meanwhile it maintains its original good properties such as conservation and GCL. Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme in terms of symmetry, non-oscillation and robustness properties.

Keywords

65M0676M20Control volume Lagrangian schemespherical symmetry preservationconservativecell-centeredcompressible flowcylindrical coordinates
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 4 , April 2012 , pp. 1144 - 1168
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]Benson, D. J., Computational methods in Lagrangian and Eulerian hydrocodes, Comput. Methods Appl. Mech. Eng., 99 (1992), 235–394.CrossRefGoogle Scholar
[3]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
[4]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
[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.CrossRefGoogle Scholar
[6]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
[7]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
[8]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.CrossRefGoogle Scholar
[9]Coggeshall, S. V. and Meyer-ter-Vehn, J., Group invariant solutions and optimal systems for multidimensional hydrodynamics, J. Math. Phys., 33 (1992), 3585–3601.Google Scholar
[10]Després, B. and Mazeran, C., Lagrangian gas dynamics in two dimensions and Lagrangian systems, Arch. Rational Mech. Anal., 178 (2005), 327–372.Google Scholar
[11]Kidder, R. E., Laser-driven compression of hollow shells: power requirements and stability limitations, Nuclear Fusion, 1 (1976), 3–14.Google Scholar
[12]Lazarus, R., Self-similar solutions for converging shocks and collapsing cavities, SIAM J. Numer. Anal., 18 (1981), 316–371.Google Scholar
[13]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.CrossRefGoogle Scholar
[14]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
[15]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.Google Scholar
[16]Margolin, L. G. and Shashkov, M. J., Using a curvilinear grid to construct symmetry-preserving discretizations for Lagrangian gas dynamics, J. Comput. Phys., 149 (1999), 389–417.Google Scholar
[17]Munz, C. D., On Godunov-type schemes for Lagrangian gas dynamics, SIAM J. Numer. Anal., 31 (1994), 17–42.Google Scholar
[18]von Neumann, J. and Richtmyer, R. D., A method for the calculation of hydrodynamics shocks, J. Appl. Phys., 21 (1950), 232–237.Google Scholar
[19]Noh, W. F., Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux, J. Comput. Phys., 72 (1987), 78–120.Google Scholar
[20]Schulz, W. D., Two-dimensional Lagrangian hydrodynamic difference equations, Methods Comput. Phys., 3 (1964), 1–45.Google Scholar
[21]Sedov, L. I., Similarity and Dimensional Methods in Mechanics, Academic Press, New York, 1959.Google Scholar
[22]Solov’ev, A. and Shashkov, M., Difference scheme for the Dirichlet particle method in cylindrical method in cylindrical coordinates, conserving symmetry of gas-dynamical flow, Differential Equations, 24 (1988), 817–823.Google Scholar
[23]Wilkins, M. L., Calculation of elastic plastic flow, Methods Comput. Phys., 3 (1964), 211–263.Google Scholar