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Robust Finite Volume Schemes for Two-Fluid Plasma Equations

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Abstract

Two-fluid plasma equations are derived by taking moments of Boltzmann equations. Ignoring collisions and viscous terms and assuming local thermodynamic equilibrium we get five moment equations for each species (electrons and ions), known as two-fluid plasma equations. These equations allow different temperatures and velocities for electrons and ions, unlike ideal magnetohydrodynamics equations. In this article, we present robust second order MUSCL schemes for two-fluid plasma equations based on Strang splitting of the flux and source terms. The source is treated both explicitly and implicitly. These schemes are shown to preserve positivity of the pressure and density. In the case of explicit treatment of source term, we derive explicit condition on the time step for it to be positivity preserving. The implicit treatment of the source term is shown to preserve positivity, unconditionally. Numerical experiments are presented to demonstrate the robustness and efficiency of these schemes.

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References

  1. Batten, P., Clarke, N., Lambert, C., Causon, M.: On the choice of wavespeeds for the HLLC Riemann solver. SIAM J. Sci. Comput. 18, 15–53 (1997)

    Article  MathSciNet  Google Scholar 

  2. Berthon, C.: Stability of the MUSCL schemes for the Euler equations. Commun. Math. Sci. 3(2), 133–157 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berthon, C.: Why the MUSCL-hancock scheme is L1-stable. Numer. Math. 104(1), 27–46 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bouchut, F.: Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Frontiers in Mathematics. Birkhauser, Basel (2004)

    Book  Google Scholar 

  5. Baboolal, S.: Finite-difference modeling of solitons induced by a density hump in a plasma multi-fluid. Math. Comput. Simul. 55, 309–316 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Baboolal, S.: High-resolution numerical simulation of 2D nonlinear wave structures in electromagnetic fluids with absorbing boundary conditions. J. Comput. Appl. Math. 234, 1710–1716 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Baboolal, S., Bharuthram, R.: Two-scale numerical solution of the electromagnetic two-fluid plasma-Maxwell equations: shock and soliton simulation. Math. Comput. Simul. 76, 3–7 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gottlieb, S., Shu, C.W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Goedbloed, H., Poedts, S.: Principles of Magnetohydrodynamics. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  10. Hakim, A., Loverich, J., Shumlak, U.: A high-resolution wave-propagation scheme for ideal two-fluid plasma equations. J. Comput. Phys. 219, 418–442 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Johnson, E.A., Rossmanith, J.A.: Collisionless magnetic reconnection in a five-moment two-fluid electron-positron plasma. In: Proceedings of Symposia in Applied Mathematics (Proceedings of HYP2008), vol. 67.2 (2009)

  12. Kumar, H., Mishra, S.: Entropy stable numerical schemes for two-fluid plasma equations. J. Sci. Comput. 52(2), 401–425 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. LeVeque, R.J.: Finite Volume Methods For Hyperbolic Problems. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  14. Loverich, J., Hakim, A., Shumlak, U.: A discontinuous Galerkin method for ideal two-fluid plasma equations. Arxiv, preprint, arXiv:1003.4542, (2010)

  15. Munz, C.D., Omnes, P., Schneider, R., Sonnendrüker, E., Voß, U.: Divergence correction techniques for Maxwell solvers based on a hyperbolic model. J. Comput. Phys. 161, 484–511 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Shumlak, U., Loverich, J.: Approximate Riemann solvers for the two-fluid plasma model. J. Comput. Phys. 187, 620–638 (2003)

    Article  MATH  Google Scholar 

  17. Shu, C.W.: TVD time discretizations. SIAM J. Math. Anal. 14, 1073–1084 (1988)

    Google Scholar 

  18. Srinivasan, B., Hakim, A., Shumlak, U.: Numerical methods for two-fluid dispersive fast MHD phenomena. Commun. Comput. Phys. 10, 183–215 (2011)

    MathSciNet  Google Scholar 

  19. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction, 2nd edn. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  20. Van Leer, B.: Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136 (1979)

    Article  Google Scholar 

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Acknowledgments

R. Abgrall has been funded in part by EU ERC Advanced Grant “ADDECCO” #226616. H. Kumar has been funded by EU ERC Advanced grant “ADDECCO” #226616 during this work.

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Authors and Affiliations

  1. BACCHUS Team, INRIA, Bordeaux, France

    Remi Abgrall

  2. Department of Mathematics, IIT Delhi, New Delhi, India

    Harish Kumar

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  1. Remi Abgrall
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  2. Harish Kumar
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Correspondence to Harish Kumar.

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Abgrall, R., Kumar, H. Robust Finite Volume Schemes for Two-Fluid Plasma Equations. J Sci Comput 60, 584–611 (2014). https://doi.org/10.1007/s10915-013-9809-6

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  • DOI: https://doi.org/10.1007/s10915-013-9809-6

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