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A Runge Kutta Discontinuous Galerkin Method for Lagrangian Compressible Euler Equations in Two-Dimensions

Published online by Cambridge University Press:  03 June 2015

Zhenzhen Li ,
Xijun Yu ,
Jiang Zhu  and
Zupeng Jia
Zhenzhen Li*
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P.R. China Graduate School, China Academy of Engineering Physics, Beijing 100088, P.R. China
Xijun Yu*
Affiliation:
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, P.R. China
Jiang Zhu*
Affiliation:
National Laboratory for Scientific Computing, LNCC/MCTI, Avenida Getúlio Vargas 333, 25651-075 Petrópolis, RJ, Brazil
Zupeng Jia*
Affiliation:
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, P.R. China
*
Email:[email protected]
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
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Abstract

This paper presents a new Lagrangian type scheme for solving the Euler equations of compressible gas dynamics. In this new scheme the system of equations is discretized by Runge-Kutta Discontinuous Galerkin (RKDG) method, and the mesh moves with the fluid flow. The scheme is conservative for the mass, momentum and total energy and maintains second-order accuracy. The scheme avoids solving the geometrical part and has free parameters. Results of some numerical tests are presented to demonstrate the accuracy and the non-oscillatory property of the scheme.

Keywords

65M6076M10Lagrangian type schemecompressible Euler equationsRKDG methodconservative scheme
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 4 , April 2014 , pp. 1184 - 1206
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Caramana, E. J., Burton, D. E. and Shashkov, M. J.et al., The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, J. Comput. Phys., 146 (1998), 227–262.Google Scholar
[2]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
[3]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
[4]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
[5]Cheng, J. and Shu, C. W., A third order conservative Lagrangian type scheme on curvilinear meshes for the compressible Euler equations, Commun. Comput. Phys., 4 (2008), 1008–1024.Google Scholar
[6]Maire, P. H., Abgrall, R. and Breil, J.et al., A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, SIAM J. Sci. Comput., 29 (2007), 1781–1824.CrossRefGoogle Scholar
[7]Maire, P. H. and Breil, J., A second-order cell-centered Lagrangian scheme for two-dimensional compressible flow problems, Int. J. Numer. Meth. Fluids, 56 (2007), 1417–1423.Google Scholar
[8]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
[9]Hui, W. H., Li, P. Y. and Li, Z. W., A unified coordinate system for solving the two-dimensional Euler equations, J. Comput. Phys., 153 (1999), 596–637.Google Scholar
[10]Jia, Z. P. and Zhang, S. D., A new high-order discontinuous Galerkin spectral finite element method for Lagrangian gas dynamics in two dimensions, J. Comput. Phys., 230 (2011), 2496–2522.CrossRefGoogle Scholar
[11]Loubere, R., Ovadia, J. and Abgrall, R., A Lagrangian discontinuous Galerkin-type method on unstructed meshes to solve hydrodynamics problems, Int. J. Numer. Meth. Fluids, 44 (2004), 645–663.CrossRefGoogle Scholar
[12]Zhao, G. Z., Variational iteration method and RKDG finite element method used in La-grangian coordinate, Ph.D. Thesis, China Academy of Engineering Physics, 2011.Google Scholar
[13]Luo, H., Baum, J. D. and Löhner, R., A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids, J. Comput. Phys., 227 (2008), 8875–8893.Google Scholar
[14]Luo, H., Baum, J. D. and Löhner, R., On the computation of multi-material flows using ALE formulation, J. Comput. Phys., 194 (2004), 304–328.Google Scholar
[15]Cockburn, B. and Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework, Math. Comp., 52 (1989), 411–435.Google Scholar
[16]Cockburn, B., Lin, S. Y. and Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems, J. Comput. Phys., 84 (1989), 90–113.Google Scholar
[17]Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics, Berlin, Springer-Verlag, 1999.CrossRefGoogle Scholar
[18]Dukowicz, J. K., Meltz, B. J. A., Vorticity errors in multi-dimensional Lagrangian codes, J. Comput. Phys., 99 (1992), 115–134.Google Scholar
[19]Després, B. and Mazeran, C., Lagrangian gas dynamics in two-dimensions and lagrangian systems, Arch. Ration. Mech. Anal., 178 (2005), 327–372.Google Scholar
[20]Landau, L. and Lifchitz, E., M’ecanique des fluides, Mir, Moscow, 1989.Google Scholar
[21]Cockburn, B. and Shu, C. W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems, J. Comput. Phys., 141 (1998), 199–224.CrossRefGoogle Scholar
[22]Jia, Z. P. and Yu, X. J., A finite volume ALE method based on approximate Riemann solution, Chinese J. Comput. Phys., 24 (2007), 543–549.Google Scholar
[23]Wilkins, M. L., Calculation of elastic-plastic flow, in Methods in Computational Physics, Vol. 3, Academic Press, New York, 1964, pp. 211–263.Google Scholar
[24]Vilar, F., Cell-centered discontinuous Galerkin discretization for two-dimensional La-grangian hydrodynamics, Comput. Fluids, 64 (2012), 64–73.Google Scholar
[25]Maire, P. H., A unified sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids, Int. J. Numer. Meth. Fluids, 65 (2011), 1281–1294.CrossRefGoogle Scholar
[26]Maire, P. H., A high-order one-step sub-cell force-based discretization for cell-centered La-grangian hydrodynamics on polygonal grids, Comput. Fluids, 46 (2011), 341–347.Google Scholar