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MHD Modeling: Aims, Usage, Scales Assessed, Caveats, Codes

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Modelling Pulsar Wind Nebulae

Part of the book series: Astrophysics and Space Science Library ((ASSL,volume 446))

Abstract

This chapter is intended to provide a review of the current state of the art modeling in the field of computational magnetohydrodynamics (MHD). The equations of MHD are first derived starting from a kinetic description and its applicability in terms of temporal and spatial scales are discussed. The chapter then focuses on Godunov-type methods which have became widespread over last decades, owing to their ability to describe high-Mach number flows and discontinuities. These schemes lean on a conservative formulation of the equations and on the solution of the Riemann problem at cell interfaces. Such flows are often encountered in astrophysical environments such as supernovae remnants, pulsar wind nebulae and relativistic jets. Finally, advanced modeling using adaptive mesh refinement techniques is also described together with an example application.

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Notes

  1. 1.

    Note that Equation 16 in Mignone et al. (2009) contains a misprint.

References

  • Anile, A.M.: Relativistic Fluids and Magneto-Fluids: With Applications in Astrophysics and Plasma Physics. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  • Anton, L., et al.: Relativistic magnetohydrodynamics: renormalized eigenvectors and full wave decomposition Riemann solver. Astrophys. J. Suppl. Ser. 188, 1–31 (2010)

    Article  ADS  Google Scholar 

  • Balsara, D.: Total variation diminishing scheme for relativistic magnetohydrodynamics. Astrophys. J. Suppl. Ser. 132, 83–101 (2001)

    Article  ADS  Google Scholar 

  • Batten, P., Clarke, N., Lambert, C., Causon, D.M.: On the choice of wavespeeds for the HLLC Riemann solver. SIAM J. Sci. Comput. 18, 1553–157 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  • Bellan, P.M.: Fundamentals of Plasma Physics. Cambridge University Press, Cambridge (2008)

    Google Scholar 

  • Berger, M.J., Colella, P.: Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82, 67–84 (1989)

    Article  ADS  MATH  Google Scholar 

  • Boyd, T.J.M., Sanderson, J.J.: The Physics of Plasmas. Cambridge University Press, New York (2003)

    Book  MATH  Google Scholar 

  • Bryan, G.L., et al.: ENZO: an adaptive mesh refinement code for astrophysics. Astrophys. J. Suppl. Ser. 211, 52 pp. (2014)

    Google Scholar 

  • Cargo, P., Gallice, G.: Roe matrices for ideal MHD and systematic construction of roe matrices for systems of conservation laws. J. Comput. Phys. 136, 446–466 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Chiuderi, C., Velli, M.: Basics of Plasma Astrophysics. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  • Davis, S.F.: Simplified second-order godunov-type methods. SIAM J. Sci. Stat. Comput. 9(3), 445–473 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  • Del Zanna, L., Zanotti, O., Bucciantini, N., Londrillo, P.: ECHO: a Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics. Astron. Astrophys. 473, 11–30 (2007)

    Article  ADS  Google Scholar 

  • Dumbser, M., Zanotti, O.: Very high order P N P M schemes on unstructured meshes for the resistive relativistic MHD equations. J. Comput. Phys. 228, 6991–7006 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Einfeldt, B., Munz, C.D., Roe, P.L., Sjögreen, B.: On Godunov-type methods near low densities. J. Comput. Phys. 92, 273–295 (1991)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Fryxell, B., et al.: FLASH: an adaptive mesh hydrodynamics code for modeling astrophysical thermonuclear flashes. Astrophys. J. Suppl. Ser. 131, 273–334 (2000)

    Article  ADS  Google Scholar 

  • Giacomazzo, B., Rezzolla, L.: The exact solution of the Riemann problem in relativistic magnetohydrodynamics. J. Fluid Mech. 562, 223–259 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Godunov, S.K.: A finite difference method for the computation of Discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47, 357–393 (1959)

    Google Scholar 

  • Gurski, K.F.: An HLLC-type approximate Riemann solver for ideal magnetohydrodynamics. SIAM J. Sci. Comput. 25, 2165 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  • Honkkila, V., Janhunen P.: HLLC solver for ideal relativistic MHD. J. Comput. Phys. 223, 643–656 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Keppends, R., Meliani, Z., van Marle, A.J., Delmont, P., Vlasis, A., van der Holst, B.: Parallel, grid-adaptive approaches for relativistic hydro and magnetohydrodynamics. J. Comput. Phys. 231, 718–744 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Khokhlov, A.M.: Fully threaded tree algorithms for adaptive refinement fluid dynamics simulations. J. Comput. Phys. 143, 519–543 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Kim, J., Balsara, D.S.: A stable HLLC Riemann solver for relativistic magnetohydrodynamics. J. Comput. Phys. 270, 634–639 (2014)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Koldoba, A.V., Kuznetsov, O.A., Ustyugova, G.V.: An approximate Riemann solver for relativistic magnetohydrodynamics. Mon. Not. R. Astron. Soc. 333, 932–942 (2002)

    Article  ADS  Google Scholar 

  • LeVeque, R.J.: Finite volume methods for hyperbolic systems. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  • Li, S.: An HLLC Riemann solver for magneto-hydrodynamics. J. Comput. Phys. 203, 344–357 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • McCorquodale, P., Colella, P.: A high-order finite-volume method for conservation laws on locally refined grids. Commun. Appl. Math. Comput. Sci. 6, 1–25 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • Mignone, A.: A simple and accurate Riemann solver for isothermal MHD. J. Comput. Phys. 225, 1427–1441 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Mignone, A.: High-order conservative reconstruction schemes for finite volume methods in cylindrical and spherical coordinates. J. Comput. Phys. 270, 784–814 (2014)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Mignone, A., Bodo, G.: An HLLC Riemann solver for relativistic flows – II. magnetohydrodynamics. Mon. Not. R. Astron. Soc. 368, 1040–1054 (2006)

    Google Scholar 

  • Mignone, A., Massaglia, S., Bodo, G.: Astrophysical jet simulations: comparing different numerical methods. Astrophys. Space Sci. 293, 199–207 (2004)

    Article  ADS  Google Scholar 

  • Mignone, A., Bodo, G., Massaglia, S., Matsakos, T., Tesileanu, O., Zanni, C., Ferrari, A.: PLUTO: a numerical code for computational astrophysics. Astrophys. J. Suppl. Ser. 170, 228–242 (2007)

    Article  ADS  Google Scholar 

  • Mignone, A., Ugliano, M., Bodo, G.: A five-wave Harten-Lax-van Leer Riemann solver for relativistic magnetohydrodynamics. Mon. Not. R. Astron. Soc. 393, 1141–1156 (2009)

    Article  ADS  Google Scholar 

  • Mignone, A., Bodo, G., Ugliano, M.: Approximate Harten-Lax-van Leer Riemann solvers for relativistic magnetohydrodynamics. In: Vázquez-Cendón, E., et al. (eds.) Numerical Methods for Hyperbolic Equations, pp. 219–226. CRC Press, Leiden (2012)

    Chapter  Google Scholar 

  • Mignone, A., Zanni, C., Tzeferacos, P., van Straalen, B., Colella, P., Bodo, G.: The PLUTO code for adaptive mesh computations in astrophysical fluid dynamics. Astrophys. J. Suppl. Ser. 198, 31 pp. (2012)

    Google Scholar 

  • Miyoshi, T., Kusano, K.: A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics. J. Comput. Phys. 208, 315–344 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Olmi, B., Del Zanna, L., Amato, E., Bandiera, R., Bucciantini, N.: On the magnetohydrodynamic modelling of the Crab nebula radio emission. Mon. Not. R. Astron. Soc. 438, 1518–1525 (2014)

    Article  ADS  Google Scholar 

  • Olmi, B., Del Zanna, L., Amato, E., Bucciantini, N.: Constraints on particle acceleration sites in the Crab nebula from relativistic magnetohydrodynamic simulations. Mon. Not. R. Astron. Soc. 449, 3149–3159 (2015)

    Article  ADS  Google Scholar 

  • Olmi, B., Del Zanna, L., Amato, E., Bucciantini, N., Mignone, A.: Multi-D magnetohydrodynamic modelling of pulsar wind nebulae: recent progress and open questions. J. Plasma Phys. 82, 30 pp. (2016)

    Google Scholar 

  • Porth, O., Komissarov, S.S., Keppens, R.: Three-dimensional magnetohydrodynamic simulations of the Crab nebula. Mon. Not. R. Astron. Soc. 438, 278–306 (2014)

    Article  ADS  Google Scholar 

  • Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 135, 250–258 (1981)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  • Teyssier, R.: Cosmology hydrodynamics with adaptive mesh refinement. A new high resolution code called RAMSES. Astron. Astrophys. 385, 337–364 (2002)

    Google Scholar 

  • Toro, E.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  • Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4, 25–34 (1994)

    Article  ADS  MATH  Google Scholar 

  • Vaidya, B., Mignone, A., Bodo, G., Massaglia, S.: Astrophysical fluid simulations of thermally ideal gases with non-constant adiabatic index: numerical implementation. Astron. Astrophys. 580, A110 (2015)

    Article  ADS  Google Scholar 

  • van der Holst, B., et al.: A multidimensional grid-adaptive relativistic magnetofluid code. Comput. Phys. Commun. 179, 617–627 (2008).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Zanotti, O., Fambri, F., Dumbser, M.: Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement. Mon. Not. R. Astron. Soc. 452, 3010–3029 (2015)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Physics Department, Turin University, Torino, Italy

    A. Mignone

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  1. A. Mignone
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Correspondence to A. Mignone .

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Editors and Affiliations

  1. ICREA & Institute of Space Sciences, (IEEC-CSIC) Campus UAB, Barcelona, Spain

    Diego F. Torres

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Mignone, A. (2017). MHD Modeling: Aims, Usage, Scales Assessed, Caveats, Codes. In: Torres, D. (eds) Modelling Pulsar Wind Nebulae. Astrophysics and Space Science Library, vol 446. Springer, Cham. https://doi.org/10.1007/978-3-319-63031-1_9

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