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Entropy Stable Numerical Schemes for Two-Fluid Plasma Equations

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Abstract

Two-fluid ideal plasma equations are a generalized form of the ideal MHD equations in which electrons and ions are considered as separate species. The design of efficient numerical schemes for the these equations is complicated on account of their non-linear nature and the presence of stiff source terms, especially for high charge to mass ratios and for low Larmor radii. In this article, we design entropy stable finite difference schemes for the two-fluid equations by combining entropy conservative fluxes and suitable numerical diffusion operators. Furthermore, to overcome the time step restrictions imposed by the stiff source terms, we devise time-stepping routines based on implicit-explicit (IMEX)-Runge Kutta (RK) schemes. The special structure of the two-fluid plasma equations is exploited by us to design IMEX schemes in which only local (in each cell) linear equations need to be solved at each time step. Benchmark numerical experiments are presented to illustrate the robustness and accuracy of these schemes.

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References

  1. 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  MathSciNet  MATH  Google Scholar 

  2. 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  MathSciNet  MATH  Google Scholar 

  3. 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  MathSciNet  MATH  Google Scholar 

  4. Barth, T.J.: Numerical methods for gas-dynamics systems on unstructured meshes. In: Krner, D., Ohlberger, M., Rohde, C. (eds.) An Introduction to Recent Developments in Theory and Numerics of Conservation Laws. Lecture Notes in Computational Science and Engineering, vol. 5, pp. 195–285. Springer, Berlin (1999)

    Chapter  Google Scholar 

  5. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, 3rd edn. Springer, Berlin (2009)

    Google Scholar 

  6. Fjordholm, U.S., Mishra, S., Tadmor, E.: Arbitrarily High order accurate entropy stable essentially non-oscillatory schemes for system of conservation laws. Research Report N. 2011–39, Seminar für Angewandte Mathmatik ETH Zürich (2011)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

  9. 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  MathSciNet  MATH  Google Scholar 

  10. Ismail, F., Roe, P.L.: Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks. J. Comput. Phys. 228, 5410–5436 (2009)

    Article  MathSciNet  MATH  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)

    Google Scholar 

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

    Book  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Google Scholar 

  17. Tadmor, E.: The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comput. 49, 91–103 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Act. Numerica 451–512 (2004)

  19. Zhang, X., Shu, C.W.: On positivity-preserving high-order Discontinuous Galerkin schemes for Euler equations on rectangular meshes. J. Comput. Phys. 229, 8918–8934 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Seminar for Applied Mathematics (SAM), Department of Mathematics, ETH Zürich, Zürich, Switzerland

    Harish Kumar & Siddhartha Mishra

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  1. Harish Kumar
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  2. Siddhartha Mishra
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Correspondence to Harish Kumar.

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Kumar, H., Mishra, S. Entropy Stable Numerical Schemes for Two-Fluid Plasma Equations. J Sci Comput 52, 401–425 (2012). https://doi.org/10.1007/s10915-011-9554-7

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  • DOI: https://doi.org/10.1007/s10915-011-9554-7

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