×

zbMATH — the first resource for mathematics

Positivity-preserving analysis of numerical schemes for ideal magnetohydrodynamics. (English) Zbl 1391.76369

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35Q35 PDEs in connection with fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] R. Artebrant and M. Torrilhon, Increasing the accuracy in locally divergence-preserving finite volume schemes for MHD, J. Comput. Phys., 227 (2008), pp. 3405–3427. · Zbl 1329.76197
[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, Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics, J. Comput. Phys., 228 (2009), pp. 5040–5056. · Zbl 1280.76030
[4] D. S. Balsara, Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics, J. Comput. Phys., 231 (2012), pp. 7504–7517.
[5] D. S. Balsara and D. Spicer, Maintaining pressure positivity in magnetohydrodynamic simulations, J. Comput. Phys., 148 (1999), pp. 133–148. · Zbl 0930.76050
[6] 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
[7] 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
[8] 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
[9] J. U. Brackbill and D. C. Barnes, The effect of nonzero \(∇·{B}\) on the numerical solution of the magnetohydrodynamic equations, J. Comput. Phys., 35 (1980), pp. 426–430. · Zbl 0429.76079
[10] 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
[11] 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
[12] 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
[13] 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
[14] 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
[15] 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
[16] C. R. Evans and J. F. Hawley, Simulation of magnetohydrodynamic flows: A constrained transport method, Astrophys. J., 332 (1988), pp. 659–677.
[17] S. Gottlieb, On high order strong stability preserving Runge-Kutta and multi step time discretizations, J. Sci. Comput., 25 (2005), pp. 105–128. · Zbl 1203.65166
[18] 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
[19] S. Gottlieb, C.-W. Shu, and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), pp. 89–112, . · Zbl 0967.65098
[20] J.-L. Guermond and B. Popov, Fast estimation from above of the maximum wave speed in the Riemann problem for the Euler equations, J. Comput. Phys., 321 (2016), pp. 908–926. · Zbl 1349.76769
[21] 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
[22] 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
[23] 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
[24] 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
[25] 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
[26] S. Li, High order central scheme on overlapping cells for magneto-hydrodynamic flows with and without constrained transport method, J. Comput. Phys., 227 (2008), pp. 7368–7393. · Zbl 1201.76311
[27] 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
[28] P. Londrillo and L. Del Zanna, High-order upwind schemes for multidimensional magnetohydrodynamics, Astrophys. J., 530 (2000), pp. 508–524.
[29] P. Londrillo and L. Del Zanna, On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: The upwind constrained transport method, J. Comput. Phys., 195 (2004), pp. 17–48. · Zbl 1087.76074
[30] S. A. Moe, J. A. Rossmanith, and D. C. Seal, Positivity-preserving discontinuous Galerkin methods with Lax-Wendroff time discretizations, J. Sci. Comput., 71 (2017), pp. 44–70. · Zbl 06849349
[31] K. G. Powell, An Approximate Riemann Solver for Magnetohydrodynamics (that Works in More than One Dimension), Tech. report, ICASE Report 94-24, NASA, Langley, VA, 1994.
[32] 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
[33] 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.
[34] M. Torrilhon, Locally divergence-preserving upwind finite volume schemes for magnetohydrodynamic equations, SIAM J. Sci. Comput., 26 (2005), pp. 1166–1191, . · Zbl 1149.76693
[35] M. Torrilhon and M. Fey, Constraint-preserving upwind methods for multidimensional advection equations, SIAM J. Numer. Anal., 42 (2004), pp. 1694–1728, . · Zbl 1146.76621
[36] G. Tóth, The \(∇·{B}=0\) constraint in shock-capturing magnetohydrodynamics codes, J. Comput. Phys., 161 (2000), pp. 605–652. · Zbl 0980.76051
[37] K. Waagan, A positive MUSCL-Hancock scheme for ideal magnetohydrodynamics, J. Comput. Phys., 228 (2009), pp. 8609–8626. · Zbl 1287.76173
[38] K. Wu, Design of provably physical-constraint-preserving methods for general relativistic hydrodynamics, Phys. Rev. D, 95 (2017), 103001.
[39] K. Wu and C.-W. Shu, Provably positive discontinuous Galerkin methods for multidimensional ideal magnetohydrodynamics, SIAM J. Sci. Comput., submitted. · Zbl 1404.65184
[40] 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
[41] 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
[42] 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), 3.
[43] 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.
[44] 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
[45] 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
[46] Z. Xu and X. Zhang, Bound-preserving high order schemes, in Handbook of Numerical Methods for Hyperbolic Problems: Applied and Modern Issues, R. Abgrall and C.-W. Shu, eds., Handb. Numer. Anal. 18, Elsevier/North–Holland, Amsterdam, 2017, pp. 81–102.
[47] 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.
[48] X. Zhang, On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier-Stokes equations, J. Comput. Phys., 328 (2017), pp. 301–343. · Zbl 1406.65091
[49] 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
[50] 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
[51] 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
[52] 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
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.