Communications in Applied Mathematics and Computational Science

On the origin of divergence errors in MHD simulations and consequences for numerical schemes

Friedemann Kemm

Full-text: Open access

Abstract

This paper investigates the origin of divergence errors in MHD simulations. For that purpose, we introduce the concept of discrete involutions for discretized conservation laws. This is done in analogue to the concept of involutions for hyperbolic conservation laws, introduced by Dafermos. By exploring the connection between discrete involutions and resonance, especially for constrained transport like MHD, we identify the lack of positive central viscosity and the assumption of one-dimensional physics in the calculation of intercell fluxes as the main sources of divergence errors. As an example of the consequences for numerical schemes, we give a hint how to modify Roe-type schemes in order to decrease the divergence errors considerably and, thus, stabilize the scheme.

Article information

Source
Commun. Appl. Math. Comput. Sci., Volume 8, Number 1 (2013), 1-38.

Dates
Received: 20 October 2010
Revised: 14 May 2012
Accepted: 10 December 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.camcos/1513732078

Digital Object Identifier
doi:10.2140/camcos.2013.8.1

Mathematical Reviews number (MathSciNet)
MR3070021

Zentralblatt MATH identifier
1282.76199

Subjects
Primary: 76W05: Magnetohydrodynamics and electrohydrodynamics 39A12: Discrete version of topics in analysis 35L45: Initial value problems for first-order hyperbolic systems 35L65: Conservation laws 35L80: Degenerate hyperbolic equations
Secondary: 35N10: Overdetermined systems with variable coefficients 65M06: Finite difference methods 39A70: Difference operators [See also 47B39] 65Z05: Applications to physics

Keywords
involutions constraint magnetohydrodynamics plasma physics Maxwell equations divergence curl operator scheme finite differences finite volume method resonance hyperbolic PDE compressible flow

Citation

Kemm, Friedemann. On the origin of divergence errors in MHD simulations and consequences for numerical schemes. Commun. Appl. Math. Comput. Sci. 8 (2013), no. 1, 1--38. doi:10.2140/camcos.2013.8.1. https://projecteuclid.org/euclid.camcos/1513732078


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References

  • J. Balbás, 2008, personal communication at the 12th Conference on Hyperbolic Problems, University of Maryland.
  • J. Balbás and E. Tadmor, Nonoscillatory central schemes for one- and two-dimensional magnetohydrodynamics equations, II: High-order semidiscrete schemes, SIAM J. Sci. Comput. 28 (2006), no. 2, 533–560.
  • D. S. Balsara, Divergence-free adaptive mesh refinement for magnetohydrodynamics, J. Comput. Phys. 174 (2001), 614–648.
  • D. S. Balsara and D. S. Spicer, A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations, J. Comput. Phys. 149 (1999), no. 2, 270–292.
  • N. Besse and D. Kr öner, Convergence of locally divergence-free discontinuous-Galerkin methods for the induction equations of the 2D-MHD system, M2AN Math. Model. Numer. Anal. 39 (2005), no. 6, 1177–1202.
  • J. U. Brackbill and D. C. Barnes, The effect of nonzero $\nabla \cdot {\bf B}$ on the numerical solution of the magnetohydrodynamic equations, J. Comput. Phys. 35 (1980), no. 3, 426–430.
  • M. Breuss, The correct use of the Lax–Friedrichs method, M2AN Math. Model. Numer. Anal. 38 (2004), no. 3, 519–540.
  • R. K. Crockett, P. Colella, R. T. Fisher, R. J. Klein, and C. I. McKee, An unsplit, cell-centered Godunov method for ideal MHD, J. Comput. Phys. 203 (2005), no. 2, 422–448.
  • C. M. Dafermos, Quasilinear hyperbolic systems with involutions, Arch. Rational Mech. Anal. 94 (1986), no. 4, 373–389.
  • ––––, Hyperbolic conservation laws in continuum physics, Grundlehren der Math. Wissenschaften, no. 325, Springer, Berlin, 2000.
  • H. De Sterck, Multi-dimensional upwind constrained transport on unstructured grids for “shallow water” magnetohydrodynamics, 15th AIAA Computational Fluid Dynamics Conference, AIAA, no. 2001-2623, 2001.
  • A. Dedner, F. Kemm, D. Kr öner, C.-D. Munz, T. Schnitzer, and M. Wesenberg, Hyperbolic divergence cleaning for the MHD equations, J. Comput. Phys. 175 (2002), no. 2, 645–673.
  • B. Einfeldt, On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal. 25 (1988), no. 2, 294–318.
  • C. R. Evans and J. F. Hawley, Simulation of general relativistic magnetohydrodynamic flows: A constrained transport method, Astrophys. J. 332 (1988), 659–677.
  • F. G. Fuchs, S. Mishra, and N. H. Risebro, Splitting based finite volume schemes for ideal MHD equations, J. Comput. Phys. 228 (2009), no. 3, 641–660.
  • F. G. Fuchs, K. H. Karlsen, S. Mishra, and N. H. Risebro, Stable upwind schemes for the magnetic induction equation, M2AN Math. Model. Numer. Anal. 43 (2009), no. 5, 825–852.
  • P. A. Gilman, Magnetohydrodynamic “shallow water” equations for the solar tachocline, Astrophys. J. Lett. 544 (2000), no. 2, L79.
  • S. K. Godunov, Non-unique “blurrings” of discontinuities in solutions of quasilinear systems, Dokl. Akad. Nauk SSSR 2 (1961), 43–44, in Russian; translated in Sov. Math. Dokl. 2 (1961), 947–949.
  • ––––, The problem of a generalized solution in the theory of quasi-linear equations and in gas dynamics, Uspehi Mat. Nauk 17 (1962), no. 3, 147–158, in Russian; translated in Russ. Math. Surv. 17 (1962), no. 3, 145–156.
  • ––––, Symmetric form of the magnetohydrodynamic equation, Chislennye Metody Mekh. Sploshnoi Sredy 3 (1972), no. 1, 26–34, in Russian.
  • A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1983), no. 3, 357–393.
  • A. Harten and J. M. Hyman, Self-adjusting grid methods for one-dimensional hyperbolic conservation laws, J. Comput. Phys. 50 (1983), no. 2, 235–269.
  • A. Harten, P. D. Lax, and B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev. 25 (1983), no. 1, 35–61.
  • F. Kemm, A carbuncle free Roe-type solver for the Euler equations, Hyperbolic problems: theory, numerics, applications (S. Benzoni-Gavage et al., eds.), Springer, Berlin, 2008, pp. 601–608.
  • ––––, Discrete involutions, resonance, and the divergence problem in MHD, Hyperbolic problems: theory, numerics and applications (E. Tadmor, J.-G. Liu, and A. E. Tzavaras, eds.), Proc. Sympos. Appl. Math., no. 67, Amer. Math. Soc., Providence, RI, 2009, pp. 725–735.
  • ––––, A comparative study of TVD-limiters–-well-known limiters and an introduction of new ones, Internat. J. Numer. Methods Fluids 67 (2011), no. 4, 404–440.
  • F. Kemm, Y.-J. Lee, C.-D. Munz, and R. Schneider, Divergence cleaning in finite-volume computations for electromagnetic wave propagations, Finite volumes for complex applications, III (R. Herbin and D. Kröner, eds.), Hermes, Paris, 2002, pp. 561–568.
  • R. J. LeVeque, Finite volume methods for hyperbolic problems, Cambridge University Press, Cambridge, 2002.
  • R. J. LeVeque et al., Clawpack \normalfont(conservation laws package), software.
  • B. Marder, A method incorporating Gauß' law into electromagnetic pic codes, J. Comput. Phys. 68 (1987), 48–55.
  • S. Mishra and E. Tadmor, Constraint preserving schemes using potential-based fluxes, I: Multidimensional transport equations, Commun. Comput. Phys. 9 (2011), no. 3, 688–710.
  • ––––, Constraint preserving schemes using potential-based fluxes, II: Genuinely multidimensional systems of conservation laws, SIAM J. Numer. Anal. 49 (2011), no. 3, 1023–1045.
  • ––––, Constraint preserving schemes using potential-based fluxes, III: Genuinely multi-dimensional schemes for the MHD equations, ESAIM Math. Model. Numer. Anal. 46 (2012), 661–680.
  • C.-D. Munz, P. Omnes, R. Schneider, E. Sonnendrücker, and U. Voß, Divergence correction techniques for Maxwell solvers based on a hyperbolic model, J. Comput. Phys. 161 (2000), no. 2, 484–511.
  • C.-D. Munz, R. Schneider, and U. Voß, A finite-volume method for the Maxwell equations in the time domain, SIAM J. Sci. Comput. 22 (2000), no. 2, 449–475.
  • C.-D. Munz, R. Schneider, E. Sonnendrücker, and U. Voss, Maxwell's equations when the charge conservation is not satisfied, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 5, 431–436.
  • H. Nessyahu and E. Tadmor, Nonoscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys. 87 (1990), no. 2, 408–463.
  • K. G. Powell, P. L. Roe, R. S. Myong, T. Gombosi, and D. de Zeeuw, An upwind scheme for magnetohydrodynamics, Workshop Méthodes numériques pour la M.H.D.
  • K. G. Powell, P. L. Roe, T. J. Linde, T. I. Gombosi, and D. L. De Zeeuw, A solution-adaptive upwind scheme for ideal magnetohydrodynamics, J. Comput. Phys. 154 (1999), no. 2, 284–309.
  • J. A. Rossmanith, An unstaggered, high-resolution constrained transport method for magnetohydrodynamic flows, SIAM J. Sci. Comput. 28 (2006), no. 5, 1766–1797.
  • M. Torrilhon and M. Fey, Constraint-preserving upwind methods for multidimensional advection equations, SIAM J. Numer. Anal. 42 (2004), no. 4, 1694–1728.
  • M. Torrilhon, Zur Numerik der idealen Magnetohydrodynamik, Ph.D. thesis, ETH Zürich, 2003.
  • ––––, Locally divergence-preserving upwind finite volume schemes for magnetohydrodynamic equations, SIAM J. Sci. Comput. 26 (2005), no. 4, 1166–1191.
  • G. Tóth, The $\nabla\cdot B=0$ constraint in shock-capturing magnetohydrodynamics codes, J. Comput. Phys. 161 (2000), no. 2, 605–652.
  • K. Waagan, A positive MUSCL-Hancock scheme for ideal magnetohydrodynamics, J. Comput. Phys. 228 (2009), no. 23, 8609–8626.
  • A. L. Zachary, A. Malagoli, and P. Colella, A higher-order Godunov method for multidimensional ideal magnetohydrodynamics, SIAM J. Sci. Comput. 15 (1994), no. 2, 263–284.