ORIGINAL PAPERS

A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations

Expand
  • School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, China

Received date: 2022-08-01

  Revised date: 2022-11-30

  Online published: 2024-04-16

Supported by

Research supported by the NSFC Grant 11901555,12271499 and the Cyrus Tang Foundation. Research supported by the NSFC Grant 11871448 and 12126604.

Abstract

In this paper, we construct a high-order discontinuous Galerkin (DG) method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics (VRMHD). To control the divergence error in the magnetic field, both the local divergence-free basis and the Godunov source term would be employed for the multi-dimensional VRMHD. Rigorous theoretical analyses are presented for one-dimensional and multi-dimensional DG schemes, respectively, showing that the scheme can maintain the positivity-preserving (PP) property under some CFL conditions when combined with the strong-stability-preserving time discretization. Then, general frameworks are established to construct the PP limiter for arbitrary order of accuracy DG schemes. Numerical tests demonstrate the effectiveness of the proposed schemes.

Cite this article

Mengjiao Jiao, Yan Jiang, Mengping Zhang . A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations[J]. Communications on Applied Mathematics and Computation, 2024 , 6(1) : 279 -310 . DOI: 10.1007/s42967-022-00247-5

References

[1] Balsara, D.S.:Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction. Astrophys. J. Suppl. Ser. 151(1), 149 (2004)
[2] Balsara, D.S.:Self-adjusting, positivity preserving high order schemes for hydrodynamics and magnetohydrodynamics. J. Comput. Phys. 231(22), 7504-7517 (2012)
[3] 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(2), 270-292 (1999)
[4] Barth, T.:On the role of involutions in the discontinuous Galerkin discretization of Maxwell and magnetohydrodynamic systems. In:Arnold, D.N., Bochev, P.B., Lehoucq, R.B., Nicolaides, R.A., Shashkov, M. (eds) Compatible Spatial Discretizations. The IMA Volumes in Mathematics and Its Applications, vol. 142, pp. 69-88. Springer, New York, NY (2006)
[5] Barth, T.J.:Numerical methods for gasdynamic systems on unstructured meshes. In:Kröner, D., Ohlberger, M., Rohde, C. (eds) An Introduction to Recent Developments in Theory and Numerics for Conservation Laws. Lecture Notes in Computational Science and Engineering, vol. 5, pp. 195-285. Springer, Berlin, Heidelberg (1999)
[6] Brackbill, J.U., Barnes, D.C.:The effect of nonzero ▽·B on the numerical solution of the magnetohydrodynamic equations. J. Comput. Phys. 35(3), 426-430 (1980)
[7] Chandrashekar, P., Gallego-Valencia, J.P., Klingenberg, C.:A Runge-Kutta discontinuous Galerkin scheme for the ideal magnetohydrodynamical model. In:Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems I. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol. 236, pp. 335-344. Springer, Cham (2016)
[8] Chandrashekar, P., Klingenberg, C.:Entropy stable finite volume scheme for ideal compressible MHD on 2-D Cartesian meshes. SIAM J. Numer. Anal. 54(2), 1313-1340 (2016)
[9] Cheng, Y., Li, F., Qiu, J., Xu, L.W.:Positivity-preserving DG and central DG methods for ideal MHD equations. J. Comput. Phys. 238, 255-280 (2013)
[10] Christlieb, A.J., Feng, X., Jiang, Y., Tang, Q.:A high-order finite difference WENO scheme for ideal magnetohydrodynamics on curvilinear meshes. SIAM J. Sci. Comput. 40(4), A2631-A2666 (2018)
[11] Christlieb, A.J., Liu, Y., Tang, Q., Xu, Z.F.:Positivity-preserving finite difference weighted ENO schemes with constrained transport for ideal magnetohydrodynamic equations. SIAM J. Sci. Comput. 37(4), A1825-A1845 (2015)
[12] Christlieb, A.J., Rossmanith, J.A., Tang, Q.:Finite difference weighted essentially non-oscillatory schemes with constrained transport for ideal magnetohydrodynamics. J. Comput. Phys. 268, 302-325 (2014)
[13] Dai, W., Woodward, P.R.:A simple finite difference scheme for multidimensional magnetohydrodynamical equations. J. Comput. Phys. 142(2), 331-369 (1998)
[14] Dedner, A., Kemm, F., Króner, D., Munz, C.D., Schnitzer, T., Wesenberg, M.:Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys. 175(2), 645-673 (2002)
[15] Evans, C.R., Hawley, J.F.:Simulation of magnetohydrodynamic flows:a constrained transport method. Astrophys. J. 332, 659-677 (1988)
[16] Godunov, S.K.:Symmetric form of the equations of magnetohydrodynamics. Numer. Methods Mech. Cont. Medium. 1, 26-34 (1972)
[17] Hu, X.Y., Adams, N.A., Shu, C.-W.:Positivity-preserving method for high-order conservative schemes solving compressible Euler equations. J. Comput. Phys. 242, 169-180 (2013)
[18] Li, F., Shu, C.-W.:Locally divergence-free discontinuous Galerkin methods for MHD equations. J. Sci. Comput. 22(1), 413-442 (2005)
[19] Li, F., Xu, L.W.:Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations. J. Comput. Phys. 231(6), 2655-2675 (2012)
[20] Li, F., Xu, L.W., Yakovlev, S.:Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field. J. Comput. Phys. 230(12), 4828-4847 (2011)
[21] Liang, C., Xu, Z.:Parametrized maximum principle preserving flux limiters for high order schemes solving multi-dimensional scalar hyperbolic conservation laws. J. Sci. Comput. 58(1), 41-60 (2014)
[22] Liu, Y., Shu, C.-W., Zhang, M.:Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes. J. Comput. Phys. 354, 163-178 (2018)
[23] Londrillo, P., Del Zanna, L.:High-order upwind schemes for multidimensional magnetohydrodynamics. Astrophys. J. 530(1), 508 (2000)
[24] Powell, K.G.:An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension). In:Hussaini, M.Y., van Leer, B., Van Rosendale, J. (eds) Upwind and High-Resolution Schemes. Springer, Berlin, Heidelberg (1997)
[25] Powell, K.G., Roe, P., Myong, R., Gombosi, T.:An upwind scheme for magnetohydrodynamics. In:AIAA 12thComputational Fluid Dynamics Conference, San Diego, CA, June 19-22 1995, AIAA-95-1704-CP, AIAA (1995)
[26] Qiu, J., Shu, C.-W.:Runge-Kutta discontinuous Galerkin method using WENO limiters. SIAM J. Sci. Comput. 26(3), 907-929 (2005)
[27] Rossmanith, J.A.:An unstaggered, high-resolution constrained transport method for magnetohydrodynamic flows. SIAM J. Sci. Comput. 28(5), 1766-1797 (2006)
[28] Ryu, D., Miniati, F., Jones, T.W., Frank, A.:A divergence-free upwind code for multidimensional magnetohydrodynamic flows. Astrophys. J. 509(1), 244-255 (1998)
[29] Torrilhon, M.:Locally divergence-preserving upwind finite volume schemes for magnetohydrodynamic equations. SIAM J. Sci. Comput. 26(4), 1166-1191 (2005)
[30] Tóth, G.:The ▽·B=0 constraint in shock-capturing magnetohydrodynamics codes. J. Comput. Phys. 161(2), 605-652 (2000)
[31] Warburton, T.C., Karniadakis, G.E.:A discontinuous Galerkin method for the viscous MHD equations. J. Comput. Phys. 152(2), 608-641 (1999)
[32] Wu, K.L.:Design of provably physical-constraint-preserving methods for general relativistic hydrodynamics. Phys. Rev. D 95(10), 103001 (2017)
[33] Wu, K.L.:Positivity-preserving analysis of numerical schemes for ideal magnetohydrodynamics. SIAM J. Numer. Anal. 56(4), 2124-2147 (2018)
[34] Wu, K.L., Jiang, H., Shu, C.-W.:Provably positive central DG schemes via geometric quasilinearization for ideal MHD equations. arXiv:2203.14853 (2022)
[35] Wu, K.L., Shu, C.-W.:A provably positive discontinuous Galerkin method for multidimensional ideal magnetohydrodynamics. SIAM J. Sci. Comput. 40(5), B1302-B1329 (2018)
[36] Wu, K.L., Shu, C.-W.:Provably positive high-order schemes for ideal magnetohydrodynamics:analysis on general meshes. Numer. Math. 142(4), 995-1047 (2019)
[37] Wu, K.L., Shu, C.-W.:Geometric quasilinearization framework for analysis and design of bound-preserving schemes. arXiv:2111.04722 (2021)
[38] Wu, K.L., Shu, C.-W.:Provably physical-constraint-preserving discontinuous Galerkin methods for multidimensional relativistic MHD equations. Numer. Math. 148(3), 699-741 (2021)
[39] Wu, K.L., Tang, H.:Admissible states and physical-constraints-preserving schemes for relativistic magnetohydrodynamic equations. Math. Models Methods Appl. Sci. 27(10), 1871-1928 (2017)
[40] Xing, Y., Zhang, X., Shu, C.-W.:Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Adv. Water Resour. 33(12), 1476-1493 (2010)
[41] Xu, Z.:Parametrized maximum principle preserving flux limiters for high order schemes solving hyperbolic conservation laws:one-dimensional scalar problem. Math. Comput. 83(289), 2213-2238 (2014)
[42] Xu, Z., Zhang, X.:Bound-preserving high-order schemes. In:Abgrall, R., Shu, C.-W. (eds) Handbook of Numerical Analysis, vol. 18, pp. 81-102. Elsevier, North-Holland, Amsterdam (2017)
[43] Yakovlev, S., Xu, L.W., Li, F.:Locally divergence-free central discontinuous Galerkin methods for ideal MHD equations. J. Comput. Sci. 4(1/2), 80-91 (2013)
[44] Yu, Y.Z., Jiang, Y., Zhang, M.:Free-stream preserving finite difference schemes for ideal magnetohydrodynamics on curvilinear meshes. J. Sci. Comput. 82(1), 1-26 (2020)
[45] Zhang, X.:On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier-Stokes equations. J. Comput. Phys. 328, 301-343 (2017)
[46] Zhang, X., Shu, C.-W.:On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229(9), 3091-3120 (2010)
[47] Zhang, X., Shu, C.-W.:On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229(23), 8918-8934 (2010)
[48] Zhang, X., Shu, C.-W.:Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws:survey and new developments. Proc. R. Soc. A Math. Phys. Eng. Sci. 467(2134), 2752-2776 (2011)
[49] Zhao, J., Tang, H.:Runge-Kutta discontinuous Galerkin methods for the special relativistic magnetohydrodynamics. J. Comput. Phys. 343, 33-72 (2017)
Options
Outlines

/