×

zbMATH — the first resource for mathematics

Extended generalized Lagrangian multipliers for magnetohydrodynamics using adaptive multiresolution methods. (English. French summary) Zbl 1329.76202
Summary: We present a new adaptive multiresoltion method for the numerical simulation of ideal magnetohydrodynamics. The governing equations, i.e., the compressible Euler equations coupled with the Maxwell equations are discretized using a finite volume scheme on a two-dimensional Cartesian mesh. Adaptivity in space is obtained via Harten’s cell average multiresolution analysis, which allows the reliable introduction of a locally refined mesh while controlling the error. The explicit time discretization uses a compact Runge-Kutta method for local time stepping and an embedded Runge-Kutta scheme for automatic time step control. An extended generalized Lagrangian multiplier approach with the mixed hyperbolic-parabolic correction type is used to control the incompressibility of the magnetic field. Applications to a two-dimensional problem illustrate the properties of the method. Memory savings and numerical divergences of magnetic field are reported and the accuracy of the adaptive computations is assessed by comparing with the available exact solution.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76W05 Magnetohydrodynamics and electrohydrodynamics
PDF BibTeX Cite
Full Text: DOI
References:
[1] F. Assous, P. Degond, E. Heintze, P.A. Raviart, and J. Segre. On a finite-element method for solving the three-dimensional Maxwell equations. J. Comput. Phys., 109(2):222–237, 1993. · Zbl 0795.65087
[2] Dinshaw S. Balsara. Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics. J. Comput. Phys., 228:5040–5056, 2009. · Zbl 1280.76030
[3] J. U. Brackbill and D. C. Barnes. Note: The effect of nonzero \nabla {\(\cdot\)} B on the numerical solution of the magnetohydrodynamic equations. J. Comput. Phys., 35(3):426–430, 1980. · Zbl 0429.76079
[4] A. Cohen, S. M. Kaber, S. M\"{}uller, and M. Postel. Fully adaptive multiresolution finite volume schemes for conservation laws. Math. Comput., 72(241):183–225, 2003. · Zbl 1010.65035
[5] A. Dedner, F. Kemm, D. Kr\"{}oner, C.-D. Munz, T.Schnitzer, and M. Wesenberg. Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys., 175:645–673, 2002. · Zbl 1059.76040
[6] A. Dedner, C. Rohde, and M. Wesenberg. A new approach to divergence cleaning in magnetohydrodynamic simulations. In ThomasY. Hou and Eitan Tadmor, editors, Hyperbolic Problems: Theory, Numerics, Applications, pages 509–518. Springer Berlin Heidelberg, 2003. · Zbl 1061.76036
[7] B. Di Pierro. M\'{}ethode d’annulation de la divergence pour les EDP hyperboliques application aux \'{}equations de la magn\'{}etohydrodynamique. Final project Master Course, Universit\'{}e de Provence, Marseille, France, 2009. (unpublished, in French).
[8] M. O. Domingues, S. M. Gomes, O. Roussel, and K. Schneider. An adaptive multiresolution scheme with local time stepping for evolutionary PDEs. J. Comput. Phys., 227(8):3758–3780, 2008. · Zbl 1139.65060
[9] M. O. Domingues, S. M. Gomes, O. Roussel, and K. Schneider. Space-time adaptive multiresolution methods for hyperbolic conservation laws: Applications to compressible Euler equations. Appl. Numer. Math., 59:2303–2311, 2009. · Zbl 1165.76031
[10] M. O. Domingues, S. M. Gomes, O. Roussel, and K. Schneider. Adaptive multiresolution methods. ESAIM Proc., 34:1–96, 2011. · Zbl 1302.65185
[11] M. O. Domingues, O. Roussel, and K. Schneider. An adaptive multiresolution method for parabolic PDEs with time-step control. Int. J. Numer. Meth. Engng., 78:652–670, 2009. · Zbl 1183.76816
[12] A. K. F. Gomes. An\'{}alise Multirresolu\c{}c\tilde{}ao adaptativa no contexto da resolu\c{}c\tilde{}ao num\'{}erica de um modelo de Magnetohidrodin\hat{}amica ideal. Master’s thesis, Instituto Nacional de Pesquisas Espaciais (INPE), S\tilde{}ao Jos\'{}e dos Campos, 2012-09-13 2012. (sid.inpe.br/mtc-m19/2012/08.10.15.02-TDI, http://urlib.net/8JMKD3MGP7W/3CE6FSE, in Portuguese).
[13] A. Harten. Multiresolution Algorithms for the Numerical Solution of Hyperbolic Conservation Laws. Commun. Pur. Appl. ESAIM: PROCEEDINGS107 · Zbl 0860.65078
[14] A. Harten. Multiresolution representation of data: a general framework. SIAM J. Numer. Anal., 33(3):385–394, 1996. · Zbl 0861.65130
[15] S. S. Komissarov, M. Barkov, and M. Lyutikov. Tearing instability in relativistic magnetically dominated plasmas. Mon. Not. R. Astron. Soc, 374:415–426, 2007.
[16] T. Mioshi and K. A. Kusano. A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics. J. Comput. Phys., 208:315–344, 2005. · Zbl 1114.76378
[17] S. M\"{}uller. Adaptive multiscale schemes for conservation laws, volume 27 of Lectures Notes in Computational Science and Engineering. Springer, Heidelberg, 2003. · Zbl 1016.76004
[18] C.-D. Munz, P. Ommes, R. Schneider, E. Sonnendr\"{}ucker, and U. Voss. Divergence corrections techiniques for Maxwell solvers based on a hyperbolic model. J. Comput. Phys., 161(2):484, 2000.
[19] K. G. Powell. An approximate Riemann solver for magnetohydrodynamics (That works in more than one dimension). Technical report.
[20] K. G. Powell, P. L. Roe, T. J. Linde, T. I. Gombosi, and D. L. D. Zeeuw. A solution-adaptative upwind scheme for ideal magnetohydrodynamics. J. Comput. Phys., 154:284–309, 1999. · Zbl 0952.76045
[21] O. Rousell, K. Schneider, A. Tsigulin, and H. Bockhorn. A conservative fully adaptative multiresolution algorithm for parabolic PDEs. J. Comput. Phys., 188:493–523, 2003. · Zbl 1022.65093
[22] M. Torrilhon. Uniqueness conditions for Riemann problems of ideal magnetohydrodynamics. J. Plasma Phys., 96:253–276, 2003.
[23] G. T\'{}oth. The \nabla {\(\cdot\)} B constraint in shock-capturing magnetohydrodynamics codes. J. Comput. Phys., 161:605–652, 2000. · Zbl 0980.76051
[24] T. S. Tricco and D. J. Price. Constrained hyperbolic divergence cleaning for smoothed particle magnetohydrodynamics. J. Comput. Phys., 231(21):7214–7236, 2012.
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.