{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T09:31:25Z","timestamp":1787563885443,"version":"build-2736575974"},"reference-count":43,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>The ideal magnetohydrodynamic (MHD) equations are important in modeling phenomena in a wide range of applications, including space weather, solar physics, laboratory plasmas, and astrophysical fluid flows. Numerical methods for the MHD equations must confront the challenge of producing approximate solutions that remain accurate near shock waves and that satisfy a divergence\u2010free constraint on the magnetic field. Failure to accomplish this often leads to unphysical solutions. In this paper, a high\u2010resolution wave propagation method is developed that utilizes a novel constrained transport technique to keep the magnetic field divergence\u2010free. This approach is based on directly solving the magnetic potential equation in conjunction with a new limiting strategy to obtain a nonoscillatory magnetic field. It is demonstrated in this work that an unstaggered definition of the divergence is the correct one to use in the case of wave propagation methods. Therefore, we solve the magnetic potential equation on the same grid as the MHD equations; hence the usual grid staggering that is found in constrained transport methods is eliminated. We demonstrate through truncation error analysis and direct numerical simulation that the resulting method is second order accurate in space and time for smooth solutions and nonoscillatory near shocks and other discontinuities. The resulting numerical method has been implemented as an extension to the clawpack software package and can be freely downloaded from the Web.<\/jats:p>","DOI":"10.1137\/050627022","type":"journal-article","created":{"date-parts":[[2006,10,23]],"date-time":"2006-10-23T11:30:50Z","timestamp":1161603050000},"page":"1766-1797","source":"Crossref","is-referenced-by-count":84,"title":["An Unstaggered, High\u2010Resolution Constrained Transport Method for Magnetohydrodynamic Flows"],"prefix":"10.1137","volume":"28","author":[{"given":"James A.","family":"Rossmanith","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,10,20]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2003.11.029"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2004.05.020"},{"key":"R3","unstructured":"D. S. Bale,\n                      Wave Propagation Algorithms on Curved Manifolds with Applications to Relativistic Hydrodynamics\n                      , Ph.D. thesis, University of Washington, Seattle, WA, 2002."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/S106482750139738X"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1086\/381377"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1998.6153"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.65.064037"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(80)90079-0"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1997.5773"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1998.5944"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2001.6961"},{"key":"R12","unstructured":"H. De Sterck,\n                      Multi\u2010dimensional upwind constrained transport on unstructured grids for \u201cshallow water\u201d magnetohydrodynamics\n                      , in Proceedings of the 15th AIAA Computational Fluid Dynamics Conference, AIAA 2001\u20102623, Anaheim, CA, 2001."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1086\/166684"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"M. Fey and M. Torrilhon,\n                      A constrained transport upwind scheme for divergence\u2010free advection\n                      , in Hyperbolic Problems: Theory, Numerics, and Applications, T. Y. Hou and E. Tadmor, eds., Springer, Berlin, 2003, pp. 529\u2013538.","DOI":"10.1007\/978-3-642-55711-8_49"},{"key":"R15","unstructured":"T. R. Fogarty,\n                      High\u2010Resolution Finite Volume Methods for Acoustics in a Rapidly\u2010Varying Heterogeneous Medium\n                      , Master\u2019s thesis, University of Washington, Seattle, WA, 1997."},{"key":"R16","unstructured":"T. R. Fogarty,\n                      Finite Volume Methods for Acoustics and Elasto\u2010plasticity with Damage in a Heterogeneous Medium\n                      , Ph.D. thesis, University of Washington, Seattle, WA, 2002."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1121\/1.428038"},{"key":"R18","first-page":"26","volume":"1","author":"Godunov S. K.","year":"1972","journal-title":"Numer. Methods Mech. Continuum Medium"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"T. I. Gombosi,\n                      Physics of the Space Environment\n                      , Cambridge University Press, Cambridge, UK, 1998.","DOI":"10.1017\/CBO9780511529474"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(83)90136-5"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(83)90066-9"},{"key":"R22","unstructured":"C. Helzel,\n                      Numerical Approximation of Conservation Laws with Stiff Source Terms for the Modelling of Detonation Waves\n                      , Ph.D. thesis, Otto\u2010von\u2010Guericke\u2010Universit\u00e4t Magdeburg, Magdeburg, Germany, 2000."},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827599357814"},{"key":"R24","doi-asserted-by":"crossref","unstructured":"J. D. Jackson,\n                      Classical Electrodynamics\n                      , 3rd ed., Wiley, New York, 1999.","DOI":"10.1119\/1.19136"},{"key":"R25","unstructured":"L. Lee,\n                      An Immersed Interface Method for Incompressible Navier\u2013Stokes Equations\n                      , Ph.D. thesis, University of Washington, Seattle, WA, 2002."},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/0733033"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1996.5603"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1998.6058"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1002\/fld.309"},{"key":"R30","doi-asserted-by":"crossref","unstructured":"R. J. LeVeque,\n                      Finite Volume Methods for Hyperbolic Problems\n                      , Cambridge University Press, Cambridge, UK, 2002.","DOI":"10.1017\/CBO9780511791253"},{"key":"R31","unstructured":"R. J. LeVeque,\n                      clawpack software\n                      , available from http:\/\/www.amath.washington.edu\/claw."},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1086\/308344"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2003.09.016"},{"key":"R34","unstructured":"K. G. Powell,\n                      An Approximate Riemann Solver for Magnetohydrodynamics (That Works in More Than One Dimension)\n                      , Tech. Report 94\u201024, ICASE, Langley, VA, 1994."},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1999.6299"},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1086\/306481"},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1137\/0721062"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2000.6519"},{"key":"R39","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2002.7177"},{"key":"R40","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827503426401"},{"key":"R41","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-005-9024-1"},{"key":"R42","doi-asserted-by":"publisher","DOI":"10.1109\/TAP.1966.1138693"},{"key":"R43","doi-asserted-by":"publisher","DOI":"10.1137\/0915019"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/050627022","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:55:11Z","timestamp":1787334911000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/050627022"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,1]]},"references-count":43,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["10.1137\/050627022"],"URL":"https:\/\/doi.org\/10.1137\/050627022","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,1]]}}}