# zbMATH — the first resource for mathematics

A provably positive discontinuous Galerkin method for multidimensional ideal magnetohydrodynamics. (English) Zbl 1404.65184

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35L65 Hyperbolic conservation laws 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs 76W05 Magnetohydrodynamics and electrohydrodynamics 76M10 Finite element methods applied to problems in fluid mechanics
Full Text:
##### References:
 [1] 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), pp. 533–560. · Zbl 1136.65340 [2] D. S. Balsara, Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophys. J. Suppl. Ser., 151 (2004), pp. 149–184. [3] D. S. Balsara, Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics, J. Comput. Phys., 231 (2012), pp. 7504–7517. [4] D. S. Balsara and D. Spicer, Maintaining pressure positivity in magnetohydrodynamic simulations, J. Comput. Phys., 148 (1999), pp. 133–148. · Zbl 0930.76050 [5] D. S. Balsara and D. Spicer, A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations, J. Comput. Phys., 149 (1999), pp. 270–292. · Zbl 0936.76051 [6] T. Barth, Numerical methods for gasdynamic systems on unstructured meshes, in An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, Springer, New York, 1999, pp. 195–285. · Zbl 0969.76040 [7] T. Barth, On the role of involutions in the discontinuous Galerkin discretization of Maxwell and magnetohydrodynamic systems, in Compatible Spatial Discretizations, Springer, New York, 2006, pp. 69–88. · Zbl 1135.78008 [8] F. Bouchut, C. Klingenberg, and K. Waagan, A multiwave approximate Riemann solver for ideal MHD based on relaxation. I: Theoretical framework, Numer. Math., 108 (2007), pp. 7–42. · Zbl 1126.76034 [9] F. Bouchut, C. Klingenberg, and K. Waagan, A multiwave approximate Riemann solver for ideal MHD based on relaxation II: Numerical implementation with 3 and 5 waves, Numer. Math., 115 (2010), pp. 647–679. · Zbl 1426.76338 [10] J. U. Brackbill and D. C. Barnes, The effect of nonzero $$∇ · \textbf{{B}}$$ on the numerical solution of the magnetohydrodynamic equations, J. Comput. Phys., 35 (1980), pp. 426–430. · Zbl 0429.76079 [11] P. Chandrashekar, P. Gallego, and C. Klingenberg, A Runge-Kutta Discontinuous Galerkin Scheme for the Ideal Magnetohydrodynamical Model, Springer Proc. Math. Stat. 236, C. Klingenberg and M. Westdickenberg, eds., Springer, New York, 2018. [12] P. Chandrashekar and C. Klingenberg, Entropy stable finite volume scheme for ideal compressible MHD on 2-D Cartesian meshes, SIAM J. Numer. Anal., 54 (2016), pp. 1313–1340. · Zbl 1381.76213 [13] Y. Cheng, F. Li, J. Qiu, and L. Xu, Positivity-preserving DG and central DG methods for ideal MHD equations, J. Comput. Phys., 238 (2013), pp. 255–280. · Zbl 1286.76162 [14] A. J. Christlieb, X. Feng, D. C. Seal, and Q. Tang, A high-order positivity-preserving single-stage single-step method for the ideal magnetohydrodynamic equations, J. Comput. Phys., 316 (2016), pp. 218–242. · Zbl 1349.76441 [15] A. J. Christlieb, Y. Liu, Q. Tang, and Z. Xu, Positivity-preserving finite difference weighted ENO schemes with constrained transport for ideal magnetohydrodynamic equations, SIAM J. Sci. Comput., 37 (2015), pp. A1825–A1845. · Zbl 1329.76225 [16] A. J. Christlieb, J. A. Rossmanith, and Q. Tang, Finite difference weighted essentially non-oscillatory schemes with constrained transport for ideal magnetohydrodynamics, J. Comput. Phys., 268 (2014), pp. 302–325. · Zbl 1349.76442 [17] B. Cockburn, S. Hou, and C.-W. Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case, Math. Comp., 54 (1990), pp. 545–581. · Zbl 0695.65066 [18] W. Dai and P. R. Woodward, A simple finite difference scheme for multidimensional magnetohydrodynamical equations, J. Comput. Phys., 142 (1998), pp. 331–369. · Zbl 0932.76048 [19] A. Dedner, F. Kemm, D. Kröner, C.-D. Munz, T. Schnitzer, and M. Wesenberg, Hyperbolic divergence cleaning for the MHD equations, J. Comput. Phys., 175 (2002), pp. 645–673. · Zbl 1059.76040 [20] P. J. Dellar, A note on magnetic monopoles and the one-dimensional MHD Riemann problem, J. Comput. Phys., 172 (2001), pp. 392–398. · Zbl 1065.35523 [21] C. R. Evans and J. F. Hawley, Simulation of magnetohydrodynamic flows: A constrained transport method, Astrophys. J., 332 (1988), pp. 659–677. [22] T. A. Gardiner and J. M. Stone, An unsplit Godunov method for ideal MHD via constrained transport, J. Comput. Phys., 205 (2005), pp. 509–539. · Zbl 1087.76536 [23] S. K. Godunov, Symmetric form of the equations of magnetohydrodynamics, Numer. Meth. Mech. Continuum Medium, 1 (1972), pp. 26–34. [24] S. Gottlieb, D. I. Ketcheson, and C.-W. Shu, High order strong stability preserving time discretizations, J. Sci. Comput., 38 (2009), pp. 251–289. · Zbl 1203.65135 [25] X. Y. Hu, N. A. Adams, and C.-W. Shu, Positivity-preserving method for high-order conservative schemes solving compressible Euler equations, J. Comput. Phys., 242 (2013), pp. 169–180. · Zbl 1311.76088 [26] P. Janhunen, A positive conservative method for magnetohydrodynamics based on HLL and Roe methods, J. Comput. Phys., 160 (2000), pp. 649–661. · Zbl 0967.76061 [27] L. Krivodonova, J. Xin, J.-F. Remacle, N. Chevaugeon, and J. E. Flaherty, Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws, Appl. Numer. Math., 48 (2004), pp. 323–338. · Zbl 1038.65096 [28] F. Li and C.-W. Shu, Locally divergence-free discontinuous Galerkin methods for MHD equations, J. Sci. Comput., 22 (2005), pp. 413–442. · Zbl 1123.76341 [29] F. Li and L. Xu, Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations, J. Comput. Phys., 231 (2012), pp. 2655–2675. · Zbl 1427.76135 [30] F. Li, L. Xu, and S. Yakovlev, Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field, J. Comput. Phys., 230 (2011), pp. 4828–4847. · Zbl 1416.76117 [31] C. Liang and Z. Xu, Parametrized maximum principle preserving flux limiters for high order schemes solving multi-dimensional scalar hyperbolic conservation laws, J. Sci. Comput., 58 (2014), pp. 41–60. · Zbl 1286.65102 [32] Y. Liu, C.-W. Shu, and M. Zhang, Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes, J. Comput. Phys., 354 (2018), pp. 163–178. · Zbl 1380.76162 [33] P. Londrillo and L. Del Zanna, High-order upwind schemes for multidimensional magnetohydrodynamics, Astrophys. J., 530 (2000), pp. 508–524. [34] K. G. Powell, An Approximate Riemann Solver for Magnetohydrodynamics (That Works in More Than One Dimension), ICASE Report No. 94-24, NASA, Langley, VA, 1994. [35] K. G. Powell, P. Roe, R. Myong, and T. Gombosi, An upwind scheme for magnetohydrodynamics, in Proceedings of the 12th Computational Fluid Dynamics Conference, 1995, pp. 661–679. · Zbl 0900.76344 [36] J. Qiu and C.-W. Shu, Runge–Kutta discontinuous Galerkin method using WENO limiters, SIAM J. Sci. Comput., 26 (2005), pp. 907–929. · Zbl 1077.65109 [37] J. A. Rossmanith, An unstaggered, high-resolution constrained transport method for magnetohydrodynamic flows, SIAM J. Sci. Comput., 28 (2006), pp. 1766–1797. · Zbl 1344.76092 [38] D. Ryu, F. Miniati, T. Jones, and A. Frank, A divergence-free upwind code for multidimensional magnetohydrodynamic flows, Astrophys. J., 509 (1998), pp. 244–255. [39] M. Torrilhon, Locally divergence-preserving upwind finite volume schemes for magnetohydrodynamic equations, SIAM J. Sci. Comput., 26 (2005), pp. 1166–1191. · Zbl 1149.76693 [40] G. Tóth, The $$∇ · {{B}} = 0$$ constraint in shock-capturing magnetohydrodynamics codes, J. Comput. Phys., 161 (2000), pp. 605–652. · Zbl 0980.76051 [41] K. Waagan, A positive MUSCL-Hancock scheme for ideal magnetohydrodynamics, J. Comput. Phys., 228 (2009), pp. 8609–8626. · Zbl 1287.76173 [42] K. Waagan, C. Federrath, and C. Klingenberg, A robust numerical scheme for highly compressible magnetohydrodynamics: Nonlinear stability, implementation and tests, J. Comput. Phys., 230 (2011), pp. 3331–3351. · Zbl 1316.76082 [43] K. Wu, Design of provably physical-constraint-preserving methods for general relativistic hydrodynamics, Phys. Rev. D, 95 (2017), 103001. [44] K. Wu, Positivity-preserving analysis of numerical schemes for ideal magnetohydrodynamics, SIAM J. Numer. Anal., 56 (2018), pp. 2124–2147. · Zbl 1391.76369 [45] K. Wu and H. Tang, High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics, J. Comput. Phys., 298 (2015), pp. 539–564. · Zbl 1349.76550 [46] K. Wu and H. Tang, Admissible states and physical-constraints-preserving schemes for relativistic magnetohydrodynamic equations, Math. Models Methods Appl. Sci., 27 (2017), pp. 1871–1928. · Zbl 1371.76096 [47] K. Wu and H. Tang, Physical-constraint-preserving central discontinuous Galerkin methods for special relativistic hydrodynamics with a general equation of state, Astrophys. J. Suppl. Ser., 228 (2017). [48] Y. Xing, X. Zhang, and C.-W. Shu, Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations, Adv. Water Res., 33 (2010), pp. 1476–1493. [49] T. Xiong, J.-M. Qiu, and Z. Xu, Parametrized positivity preserving flux limiters for the high order finite difference WENO scheme solving compressible Euler equations, J. Sci. Comput., 67 (2016), pp. 1066–1088. · Zbl 1383.76365 [50] Z. Xu, Parametrized maximum principle preserving flux limiters for high order schemes solving hyperbolic conservation laws: One-dimensional scalar problem, Math. Comp., 83 (2014), pp. 2213–2238. · Zbl 1300.65063 [51] Z. Xu and X. Zhang, Bound-Preserving High Order Schemes, in Handbook of Numerical Methods for Hyperbolic Problems: Applied and Modern Issues, Vol. 18, R. Abgrall and C.-W. Shu, eds., North-Holland, Amsterdam, 2017, pp. 81–102. [52] S. Yakovlev, L. Xu, and F. Li, Locally divergence-free central discontinuous Galerkin methods for ideal MHD equations, J. Comput. Sci., 4 (2013), pp. 80–91. [53] X. Zhang, On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier-Stokes equations, J. Comput. Phys., 328 (2017), pp. 301–343. [54] X. Zhang and C.-W. Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comput. Phys., 229 (2010), pp. 3091–3120. · Zbl 1187.65096 [55] X. Zhang and C.-W. Shu, On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, J. Comput. Phys., 229 (2010), pp. 8918–8934. · Zbl 1282.76128 [56] X. Zhang and C.-W. Shu, Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: Survey and new developments, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), pp. 2752–2776. · Zbl 1222.65107 [57] X. Zhang and C.-W. Shu, Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms, J. Comput. Phys., 230 (2011), pp. 1238–1248. · Zbl 1391.76375 [58] J. Zhao and H. Tang, Runge-Kutta discontinuous Galerkin methods for the special relativistic magnetohydrodynamics, J. Comput. Phys., 343 (2017), pp. 33–72. · Zbl 1380.76048
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.