×

zbMATH — the first resource for mathematics

Hyperbolic divergence cleaning for the MHD equations. (English) Zbl 1059.76040
Summary: In simulations of magnetohydrodynamic (MHD) processes, the violation of the divergence constraint causes severe stability problems. In this paper, we develop and test a new approach to the stabilization of numerical schemes. Our technique can be easily implemented in any existing code, since there is no need to modify the solver for the MHD equations. It is based on a modified system in which the divergence constraint is coupled with the conservation laws by introducing a generalized Lagrange multiplier. We suggest a formulation in which the divergence errors are transported to the domain boundaries with maximal admissible speed and are damped at the same time. This corrected system is hyperbolic, and the density, momentum, magnetic induction, and total energy density are still conserved. In comparison to results obtained without correction or with the standard “divergence source terms,” our approach seems to yield more robust schemes with significantly smaller divergence errors.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Software:
ZEUS
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aslan, N., computational investigations of ideal magnetohydrodynamic plasmas with discontinuities, (1993), Univ. Michigan
[2] Aslan, N.; Kammash, T., Developing numerical fluxes with new sonic fix for MHD equations, J. comput. phys., 133, 43, (1997) · Zbl 0883.76054
[3] Assous, F.; Degond, P.; Heintze, E.; Raviart, P.A.; Segre, J., On a finite-element method for solving the three-dimensional Maxwell equations, J. comput. phys., 109, 222, (1993) · Zbl 0795.65087
[4] Balsara, D.S., Linearized formulation of the Riemann problem for adiabatic and isothermal magnetohydrodynamics, Astrophys. J. suppl., 116, 119, (1998)
[5] Balsara, D.S., Total variation diminishing scheme for adiabatic and isothermal magnetohydrodynamics, Astrophys. J. suppl., 116, 133, (1998)
[6] Balsara, D.S.; Spicer, D.S., A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations, J. comput. phys., 149, 270, (1999) · Zbl 0936.76051
[7] Barmin, A.A.; Kulikovskiy, A.G.; Pogorelov, N.V., Shock-capturing approach and nonevolutionary solutions in magnetohydrodynamics, J. comput. phys., 126, 77, (1996) · Zbl 0863.76039
[8] T. J. Barth, Numerical methods for gasdynamic systems on unstructured meshes, in, An Introduction to Recent Developments in Theory and Numerics for Conservation Laws: Proceedings of the International School, Freiburg/Littenweiler, Germany, October 20-24, 1997, edited by, D. Kröner, M. Ohlberger, and C. Rohde, Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin, 1999, Vol, 5, p, 195.
[9] Bezard, F.; Despres, B., An entropic solver for ideal Lagrangian magnetohydrodynamics, J. comput. phys., 154, 65, (1999) · Zbl 0952.76053
[10] J. P. Boris, Relativistic plasma simulation—optimization of a hybrid code, in Proc. Fourth Conf. Num. Sim. Plasmas, Naval Res. Lab., 1971, pp. 3-67.
[11] Brackbill, J., Fluid modeling of magnetized plasmas, Space sci. rev., 42, 153, (1985)
[12] Brackbill, J.U.; Barnes, D.C., Note: the effect of nonzero ▿·B on the numerical solution of the magnetohydrodynamic equations, J. comput. phys., 35, 426, (1980) · Zbl 0429.76079
[13] Brecht, S.H.; Lyon, J.; Fedder, J.A.; Hain, K., A simulation study of east – west IMF effects on the magnetosphere, Geophys. res. lett., 8, 397, (1981)
[14] Brio, M.; Wu, C.C., An upwind differencing scheme for the equations of ideal magnetohydrodynamics, J. comput. phys., 75, 400, (1988) · Zbl 0637.76125
[15] 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, (1997) · Zbl 0919.76053
[16] Dai, W.; Woodward, P.R., An approximate Riemann solver for ideal magnetohydrodynamics, J. comput. phys., 111, 354, (1994) · Zbl 0797.76052
[17] Dai, W.; Woodward, P.R., A simple Riemann solver and high-order Godunov schemes for hyperbolic systems of conservation laws, J. comput. phys., 121, 51, (1995) · Zbl 0838.65088
[18] Dai, W.; Woodward, P.R., A simple finite difference scheme for multidimensional magnetohydrodynamical equations, J. comput. phys., 142, 331, (1998) · Zbl 0932.76048
[19] A. Dedner, C. Rohde, and, M. Wesenberg, A MHD-simulation in solar physics, in, Finite Volumes for Complex Applications. II. Problems and Perspectives, edited by, R. Vilsmeier, F. Benkhaldoun, and D. Hänel, Hermès Sci. Publ. Paris, 1999, p, 491. · Zbl 1052.65542
[20] Degond, P.; Peyrard, P.-F.; Russo, G.; Villedieux, P., Polynomial upwind schemes for hyperbolic systems, C. R. acad. sci. Paris ser. I, math., 328, 479, (1999) · Zbl 0933.65101
[21] DeVore, C.R., Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics, J. comput. phys., 92, 142, (1991) · Zbl 0716.76056
[22] Evans, C.R.; Hawley, J.F., Simulation of magnetohydrodynamic flows—a constrained transport method, Astrophys. J., 332, 659, (1988)
[23] Evans, L.C., partial differential equations, (1998), Amer. Math. Soc Providence
[24] Frank, A.; Jones, T.W.; Ryu, D.; Gaalaas, J.B., The magnetohydrodynamic kelvin – helmholtz instability: A two-dimensional numerical study, Astrophys. J., 460, 777, (1996)
[25] Godunov, S.K., The symmetric form of magnetohydrodynamics equation, Numer. methods mech. contin. media, 1, 26, (1972)
[26] Holland, R., Finite-difference solution of Maxwell’s equations in generalized nonorthogonal coordinates, IEEE trans. nucl. sci., 30, 4589, (1983)
[27] Jiang, B.-N., the least-squares finite element method: theory and applications in computational fluid dynamics and electromagnetics, (1998), Springer-Verlag Berlin/Heidelberg
[28] Jiang, B.-N.; Wu, J.; Povinelli, L.A., The origin of spurious solutions in computational electromagnetics, J. comput. phys., 125, 104, (1996) · Zbl 0848.65086
[29] Londrillo, P.; Del Zanna, L., High-order upwind schemes for multidimensional magnetohydrodynamics, Astrophys. J., 530, 508, (2000)
[30] Madsen, N.K.; Ziolkowski, R.W., A three-dimensional modified finite volume technique for Maxwell’s equations, Electromagnetics, 10, 147, (1990)
[31] Malagoli, A.; Bodo, G.; Rosner, R., On the nonlinear evolution of magnetohydrodynamic kelvin – helmholtz instabilities, Astrophys. J., 456, 708, (1996)
[32] Marder, B., A method for incorporating Gauss’ law into electromagnetic PIC codes, J. comput. phys., 68, 48, (1987) · Zbl 0603.65079
[33] C.-D. Munz, F. Kemm, R. Schneider, and, E. Sonnendrücker, Divergence corrections in the numerical simulation of electromagnetic wave propagation, in, Proceedings of the 8th International Conference on Hyperbolic Problems (Magdeburg), 2000, to appear.
[34] 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, (2000) · Zbl 0970.78010
[35] Munz, C.-D.; Schneider, R.; Sonnendrücker, E.; Voss, U., Maxwell’s equations when the charge conservation is not satisfied, C. R. acad. sci. Paris ser. I, math., 328, 431, (1999) · Zbl 0937.78005
[36] K. G. Powell, An Approximate Riemann Solver for Magnetohydrodynamics (That Works in More than One Dimension), ICASE-Report 94-24 (NASA CR-194902), NASA Langley Research Center, Hampton, VA, 23681-0001, 8. April 1994.
[37] Powell, K.G.; Roe, P.L.; Linde, T.J.; Gombosi, T.I.; De Zeeuw, D.L., A solution-adaptive upwind scheme for ideal magnetohydrodynamics, J. comput. phys., 154, 284, (1999) · Zbl 0952.76045
[38] K. G. Powell, P. L. Roe, R. S. Myong, T. Gombosi, and, D. De Zeeuw, An upwind scheme for magnetohydrodynamics, in, Numerical Methods for Fluid Dynamics, edited by, K. W. Morton and M. J. Baines, Clarendon, Oxford, 1995, Vol, V, p, 163. · Zbl 0900.76344
[39] Ryu, D.; Miniati, F.; Jones, T.W.; Frank, A., A divergence-free upwind code for multidimensional magnetohydrodynamic flows, Astrophys. J., 509, 244, (1998)
[40] Starke, G., Multilevel preconditioning for the time-harmonic Maxwell equations, 12th annual review of progress in applied computational electromagnetics, 630, (1996)
[41] Stone, J.M.; Norman, M.L., ZEUS2D: A radiation magnetohydrodynamics code for astrophysical flows in two space dimensions. II the magnetohydrodynamic algorithms and tests, Astrophys. J. suppl., 80, 791, (1992)
[42] A. Taflove, Re-inventing electromagnetics: Supercomputing solution of Maxwell’s equations via direct time integration on space grids, Paper 92-0333, AIAA, 1992.
[43] Tóth, G., The ▿·B=0 constraint in shock-capturing magnetohydrodynamics codes, J. comput. phys., 161, 605, (2000) · Zbl 0980.76051
[44] Tóth, G.; Odstrčil, D., Comparison of some flux corrected transport and total variation diminishing numerical schemes for hydrodynamic and magnetohydrodynamic problems, J. comput. phys., 128, 82, (1996) · Zbl 0860.76061
[45] M. Wesenberg, A note on MHD Riemann solvers, Preprint 35, Albert-Ludwigs-Universität, Mathematische Fakultät, Freiburg, November 2000. · Zbl 1037.76038
[46] Yee, K.S., Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE trans. antennas propag., 14, 302, (1966) · Zbl 1155.78304
[47] Zachary, A.L.; Colella, P., Note: A higher-order Godunov method for the equations of ideal magnetohydrodynamics, J. comput. phys., 99, 341, (1992) · Zbl 0744.76092
[48] Zachary, A.L.; Malagoli, A.; Colella, P., A higher-order Godunov method for multidimensional ideal magnetohydrodynamics, SIAM J. sci. comput., 15, 263, (1994) · Zbl 0797.76063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.