Skip to main content
SpringerLink
Log in

High Order Structure-Preserving Finite Difference WENO Schemes for MHD Equations with Gravitation in all Sonic Mach Numbers

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, we develop a high-order semi-implicit (SI) structure-preserving finite difference weighted essentially nonoscillatory (WENO) scheme for magnetohydrodynamic (MHD) equations with a gravitational source. The proposed scheme is well-balanced for magnetic steady states, divergence-free for the magnetic field, conservative in the high Mach regime, and exhibits asymptotic preserving (AP) and asymptotically accurate (AA) properties in the incompressible low sonic Mach regime. The constrained transport method is applied to maintain a discrete divergence-free magnetic field. The sonic Mach number \(\varepsilon \) ranging from 0 to \(\mathcal {O}(1)\) is taken into account for all Mach flows. One of the crucial and novel ingredients is the addition of an evolution equation for the perturbation of potential temperature as an auxiliary equation to the conservative MHD system. This addition ensures a correct asymptotic low sonic Mach limit and helps to effectively capture shocks in the compressible high Mach regime. A well-balanced finite difference WENO scheme is designed for conservative variables of the resulting system. With stiffly accurate SI implicit-explicit Runge–Kutta time discretizations, the AP and AA properties are formally proven. Numerical experiments are provided to validate the effectiveness and structure-preserving properties of the proposed scheme.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Code availability

The codes generated in this study are available from the corresponding author on reasonable request.

Data availability

No data sets were generated or analyzed during the current study.

References

  1. Arun, K., Samantaray, S.: Asymptotic preserving low Mach number accurate IMEX finite volume schemes for the isentropic Euler equations. J. Sci. Comput. 82(2), 1–32 (2020)

    Article  MathSciNet  Google Scholar 

  2. Aschwanden, M.: Physics of the Solar Corona: An Introduction with Problems and Solutions. Springer, Berlin (2006)

    Google Scholar 

  3. Balsara, D.S.: Total variation diminishing scheme for adiabatic and isothermal magnetohydrodynamics. Astrophys. J. Suppl. Ser. 116(1), 133 (1998)

    Article  Google Scholar 

  4. Balsara, D.S.: Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction. Astrophys. J. Suppl. Ser. 151(1), 149 (2004)

    Article  Google Scholar 

  5. 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)

    Article  MathSciNet  Google Scholar 

  6. Bannon, P.R.: On the anelastic approximation for a compressible atmosphere. J. Atmos. Sci. 53(23), 3618–3628 (1996)

    Article  MathSciNet  Google Scholar 

  7. Birke, C., Boscheri, W., Klingenberg, C.: A High Order Semi-Implicit Scheme for Ideal Magnetohydrodynamics. Springer Proceedings in Mathematics & Statistics, Finite Volume and Complex Applications X (2023)

  8. Bispen, G., Arun, K.R., Lukáčová-Medvidová, M., Noelle, S.: IMEX large time step finite volume methods for low Froude number shallow water flows. Commun. Comput. Phys. 16(2), 307–347 (2014)

    Article  MathSciNet  Google Scholar 

  9. Bispen, G., Lukáčová-Medvid’ová, M., Yelash, L.: Asymptotic preserving IMEX finite volume schemes for low Mach number Euler equations with gravitation. J. Comput. Phys. 335, 222–248 (2017)

    Article  MathSciNet  Google Scholar 

  10. Bogdan, T., Hansteen, M.C.V., McMurry, A., Rosenthal, C., Johnson, M., Petty-Powell, S., Zita, E., Stein, R., McIntosh, S., Nordlund, Å.: Waves in the magnetized solar atmosphere. II. Waves from localized sources in magnetic flux concentrations. Astrophys. J. 599(1), 626 (2003)

    Article  Google Scholar 

  11. Boscarino, S.: Error analysis of IMEX Runge–Kutta methods derived from differential-algebraic systems. SIAM J. Numer. Anal. 45(4), 1600–1621 (2007)

    Article  MathSciNet  Google Scholar 

  12. Boscarino, S., Pareschi, L., Russo, G.: Implicit-explicit Runge–Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit. SIAM J. Sci. Comput. 35(1), A22–A51 (2013)

    Article  MathSciNet  Google Scholar 

  13. Boscarino, S., Filbet, F., Russo, G.: High order semi-implicit schemes for time dependent partial differential equations. J. Sci. Comput. 68(3), 975–1001 (2016)

    Article  MathSciNet  Google Scholar 

  14. Boscarino, S., Russo, G., Scandurra, L.: All Mach number second order semi-implicit scheme for the Euler equations of gas dynamics. J. Sci. Comput. 77(2), 850–884 (2018)

    Article  MathSciNet  Google Scholar 

  15. Boscarino, S., Qiu, J.-M., Russo, G., Xiong, T.: A high order semi-implicit IMEX WENO scheme for the all-Mach isentropic Euler system. J. Comput. Phys. 392, 594–618 (2019)

    Article  MathSciNet  Google Scholar 

  16. Boscarino, S., Qiu, J., Russo, G., Xiong, T.: High order semi-implicit WENO schemes for all-Mach full Euler system of gas dynamics. SIAM J. Sci. Comput. 44(2), B368–B394 (2022)

    Article  MathSciNet  Google Scholar 

  17. Boscheri, W., Pareschi, L.: High order pressure-based semi-implicit IMEX schemes for the 3D Navier–Stokes equations at all Mach numbers. J. Comput. Phys. 434, 110206 (2021)

    Article  MathSciNet  Google Scholar 

  18. Boscheri, W., Dimarco, G., Loubère, R., Tavelli, M., Vignal, M.-H.: A second order all Mach number IMEX finite volume solver for the three dimensional Euler equations. J. Comput. Phys. 415, 109486 (2020)

    Article  MathSciNet  Google Scholar 

  19. Brackbill, J.U., Barnes, D.C.: The effect of nonzero \(\nabla \cdot B\) on the numerical solution of the magnetohydrodynamic equations. J. Comput. Phys. 35(3), 426–430 (1980)

  20. Chandrashekar, P., Klingenberg, C.: A second order well-balanced finite volume scheme for Euler equations with gravity. SIAM J. Sci. Comput. 37(3), B382–B402 (2015)

    Article  MathSciNet  Google Scholar 

  21. Chen, W., Wu, K., Xiong, T.: High order asymptotic preserving finite difference WENO schemes with constrained transport for MHD equations in all sonic Mach numbers. J. Comput. Phys. 488, 112240 (2023)

    Article  MathSciNet  Google Scholar 

  22. Cheng, B., Ju, Q., Schochet, S.: Three-scale singular limits of evolutionary PDEs. Arch. Ration. Mech. Anal. 229, 601–625 (2018)

    Article  MathSciNet  Google Scholar 

  23. Cheng, B., Ju, Q., Schochet, S.: Convergence rate estimates for the low Mach and Alfvén number three-scale singular limit of compressible ideal magnetohydrodynamics. ESAIM Math. Model. Numer. Anal. 55, S733–S759 (2021)

    Article  Google Scholar 

  24. 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)

    Article  MathSciNet  Google Scholar 

  25. Christlieb, A.J., Liu, Y., Tang, Q., Xu, Z.: Positivity-preserving finite difference weighted ENO schemes with constrained transport for ideal magnetohydrodynamic equations. SIAM J. Sci. Comput. 37(4), A1825–A1845 (2015)

    Article  MathSciNet  Google Scholar 

  26. Christlieb, A.J., Feng, X., Seal, D.C., Tang, Q.: A high-order positivity-preserving single-stage single-step method for the ideal magnetohydrodynamic equations. J. Comput. Phys. 316, 218–242 (2016)

    Article  MathSciNet  Google Scholar 

  27. 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)

    Article  MathSciNet  Google Scholar 

  28. Cordier, F., Degond, P., Kumbaro, A.: An asymptotic-preserving all-speed scheme for the Euler and Navier–Stokes equations. J. Comput. Phys. 231(17), 5685–5704 (2012)

    Article  MathSciNet  Google Scholar 

  29. Cui, W., Ou, Y., Ren, D.: Incompressible limit of full compressible magnetohydrodynamic equations with well-prepared data in 3-D bounded domains. J. Math. Anal. Appl. 427(1), 263–288 (2015)

    Article  MathSciNet  Google Scholar 

  30. Dai, W., Woodward, P.R.: On the divergence-free condition and conservation laws in numerical simulations for supersonic magnetohydrodynamical flows. Astrophys. J. 494(1), 317 (1998)

    Article  Google Scholar 

  31. Dedner, A., Kemm, F., Kröner, D., Munz, C.-D., Schnitzer, T., Wesenberg, M.: Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys. 175(2), 645–673 (2002)

    Article  MathSciNet  Google Scholar 

  32. Degond, P., Jin, S., Liu, J.: Mach-number uniform asymptotic-preserving gauge schemes for compressible flows. Bull. Inst. Math. Academia Sinica 2(4), 851 (2007)

    MathSciNet  Google Scholar 

  33. Dimarco, G., Loubère, R., Vignal, M.-H.: Study of a new asymptotic preserving scheme for the Euler system in the low Mach number limit. SIAM J. Sci. Comput. 39(5), A2099–A2128 (2017)

    Article  MathSciNet  Google Scholar 

  34. Dimarco, G., Loubère, R., Michel-Dansac, V., Vignal, M.-H.: Second-order implicit-explicit total variation diminishing schemes for the Euler system in the low Mach regime. J. Comput. Phys. 372, 178–201 (2018)

    Article  MathSciNet  Google Scholar 

  35. Dumbser, M., Balsara, D.S., Tavelli, M., Fambri, F.: A divergence-free semi-implicit finite volume scheme for ideal, viscous, and resistive magnetohydrodynamics. Int. J. Numer. Meth. Fluids 89(1–2), 16–42 (2019)

    Article  MathSciNet  Google Scholar 

  36. Duran, A., Marche, F., Turpault, R., Berthon, C.: Asymptotic preserving scheme for the shallow water equations with source terms on unstructured meshes. J. Comput. Phys. 287, 184–206 (2015)

    Article  MathSciNet  Google Scholar 

  37. Edelmann, P.V., Horst, L., Berberich, J.P., Andrassy, R., Higl, J., Leidi, G., Klingenberg, C., Röpke, F.: Well-balanced treatment of gravity in astrophysical fluid dynamics simulations at low Mach numbers. Astron. Astrophys. 652, A53 (2021)

    Article  Google Scholar 

  38. Evans, C.R., Hawley, J.F.: Simulation of magnetohydrodynamic flows: a constrained transport method. Astrophys. J. 332, 659–677 (1988)

    Article  Google Scholar 

  39. Fambri, F.: A novel structure preserving semi-implicit finite volume method for viscous and resistive magnetohydrodynamics. Int. J. Numer. Meth. Fluids 93(12), 3447–3489 (2021)

    Article  MathSciNet  Google Scholar 

  40. Fuchs, F., McMurry, A., Mishra, S., Risebro, N., Waagan, K.: Finite volume methods for wave propagation in stratified magneto-atmospheres. Commun. Comput. Phys. 7(3), 473–509 (2010)

    Article  MathSciNet  Google Scholar 

  41. Fuchs, F., McMurry, A., Mishra, S., Risebro, N., Waagan, K.: High order well-balanced finite volume schemes for simulating wave propagation in stratified magnetic atmospheres. J. Comput. Phys. 229(11), 4033–4058 (2010)

    Article  MathSciNet  Google Scholar 

  42. Fuchs, F., McMurry, A., Mishra, S., Waagan, K.: Simulating waves in the upper solar atmosphere with SURYA: a well-balanced high-order finite-volume code. Astrophys. J. 732(2), 75 (2011)

    Article  Google Scholar 

  43. Ghosh, D., Constantinescu, E.M.: Well-balanced, conservative finite difference algorithm for atmospheric flows. AIAA J. 54(4), 1370–1385 (2016)

    Article  Google Scholar 

  44. Haack, J., Jin, S., Liu, J.-G.: An all-speed asymptotic-preserving method for the isentropic Euler and Navier–Stokes equations. Commun. Comput. Phys. 12(4), 955–980 (2012)

    Article  MathSciNet  Google Scholar 

  45. Han, J., Tang, H.: An adaptive moving mesh method for two-dimensional ideal magnetohydrodynamics. J. Comput. Phys. 220(2), 791–812 (2007)

    Article  MathSciNet  Google Scholar 

  46. Helzel, C., Rossmanith, J.A., Taetz, B.: An unstaggered constrained transport method for the 3D ideal magnetohydrodynamic equations. J. Comput. Phys. 230(10), 3803–3829 (2011)

    Article  MathSciNet  Google Scholar 

  47. Huang, G., Xing, Y., Xiong, T.: High order well-balanced asymptotic preserving finite difference WENO schemes for the shallow water equations in all Froude numbers. J. Comput. Phys. 463, 111255 (2022)

    Article  MathSciNet  Google Scholar 

  48. Huang, G., Xing, Y., Xiong, T.: High order asymptotic preserving well-balanced finite difference WENO schemes for all Mach full Euler equations with gravity. Commun. Comput. Phys. (In Press). arXiv:2210.01641 (2023)

  49. Jardin, S.: Review of implicit methods for the magnetohydrodynamic description of magnetically confined plasmas. J. Comput. Phys. 231(3), 822–838 (2012)

    Article  MathSciNet  Google Scholar 

  50. Jiang, G.-S., Peng, D.: Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21(6), 2126–2143 (2000)

    Article  MathSciNet  Google Scholar 

  51. Jiang, G.-S., Wu, C.-C.: A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 150(2), 561–594 (1999)

    Article  MathSciNet  Google Scholar 

  52. Jiang, S., Ju, Q., Li, F.: Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. Commun. Math. Phys. 297(2), 371–400 (2010)

    Article  MathSciNet  Google Scholar 

  53. Jiang, S., Ju, Q., Li, F.: Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations. Nonlinearity 25(5), 1351 (2012)

    Article  MathSciNet  Google Scholar 

  54. Jiang, S., Ju, Q., Xu, X.: Small Alfvén number limit for incompressible magneto-hydrodynamics in a domain with boundaries. Sci. China Math. 62, 2229–2248 (2019)

    Article  MathSciNet  Google Scholar 

  55. Jin, S.: Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21, 441–454 (1999)

    Article  MathSciNet  Google Scholar 

  56. Jin, S.: Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review. Lecture notes for summer school on methods and models of kinetic theory (M &MKT), Porto Ercole (Grosseto, Italy), pp. 177–216 (2010)

  57. Jin, S.: Asymptotic-preserving schemes for multiscale physical problems. Acta Numer. 31, 415–489 (2022)

    Article  MathSciNet  Google Scholar 

  58. Ju, Q., Schochet, S., Xu, X.: Singular limits of the equations of compressible ideal magneto-hydrodynamics in a domain with boundaries. Asymptot. Anal. 113(3), 137–165 (2019)

    MathSciNet  Google Scholar 

  59. Kanbar, F., Touma, R., Klingenberg, C.: Well-balanced central scheme for the system of MHD equations with gravitational source term. Commun. Comput. Phys. 32(3), 878–898 (2022)

    Article  MathSciNet  Google Scholar 

  60. Käppeli, R., Mishra, S.: Well-balanced schemes for the Euler equations with gravitation. J. Comput. Phys. 259, 199–219 (2014)

    Article  MathSciNet  Google Scholar 

  61. Klein, R.: Asymptotics, structure, and integration of sound-proof atmospheric flow equations. Theoret. Comput. Fluid Dyn. 23(3), 161–195 (2009)

    Article  Google Scholar 

  62. Krause, G.: Hydrostatic equilibrium preservation in MHD numerical simulation with stratified atmospheres-explicit Godunov-type schemes with MUSCL reconstruction. Astron. Astrophys. 631, A68 (2019)

    Article  Google Scholar 

  63. Leidi, G., Birke, C., Andrassy, R., Higl, J., Edelmann, P., Wiest, G., Klingenberg, C., Röpke, F.: A finite-volume scheme for modeling compressible magnetohydrodynamic flows at low Mach numbers in stellar interiors. Astron. Astrophys. 668, A143 (2022)

    Article  Google Scholar 

  64. Li, F., Shu, C.-W.: Locally divergence-free discontinuous Galerkin methods for MHD equations. J. Sci. Comput. 22(1), 413–442 (2005)

    Article  MathSciNet  Google Scholar 

  65. Li, G., Xing, Y.: Well-balanced finite difference weighted essentially non-oscillatory schemes for the Euler equations with static gravitational fields. Comput. Math. Appl. 75(6), 2071–2085 (2018)

    Article  MathSciNet  Google Scholar 

  66. Li, F., Xu, L.: Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations. J. Comput. Phys. 231(6), 2655–2675 (2012)

    Article  MathSciNet  Google Scholar 

  67. Liu, X.: A well-balanced asymptotic preserving scheme for the two-dimensional shallow water equations over irregular bottom topography. SIAM J. Sci. Comput. 42(5), B1136–B1172 (2020)

    Article  MathSciNet  Google Scholar 

  68. Liu, X., Chertock, A., Kurganov, A.: An asymptotic preserving scheme for the two-dimensional shallow water equations with Coriolis forces. J. Comput. Phys. 391, 259–279 (2019)

    Article  MathSciNet  Google Scholar 

  69. Liu, M., Feng, X., Wang, X.: Implementation of the HLL-GRP solver for multidimensional ideal MHD simulations based on finite volume method. J. Comput. Phys. 473, 111687 (2023)

    Article  MathSciNet  Google Scholar 

  70. Luo, J., Xu, K., Liu, N.: A well-balanced symplecticity-preserving gas-kinetic scheme for hydrodynamic equations under gravitational field. SIAM J. Sci. Comput. 33(5), 2356–2381 (2011)

    Article  MathSciNet  Google Scholar 

  71. Mamashita, T., Kitamura, K., Minoshima, T.: SLAU2-HLLD numerical flux with wiggle-sensor for stable low Mach magnetohydrodynamics simulations. Comput. Fluids 231, 105165 (2021)

    Article  MathSciNet  Google Scholar 

  72. Matthaeus, W.H., Brown, M.R.: Nearly incompressible magnetohydrodynamics at low Mach number. Phys. Fluids 31(12), 3634–3644 (1988)

    Article  MathSciNet  Google Scholar 

  73. Minoshima, T., Miyoshi, T.: A low-dissipation HLLD approximate Riemann solver for a very wide range of Mach numbers. J. Comput. Phys. 446, 110639 (2021)

    Article  MathSciNet  Google Scholar 

  74. Noelle, S., Bispen, G., Arun, K.R., Lukáčová-Me\(\acute{\text{d}}\)vidová, M., Munz, C.-D.: A weakly asymptotic preserving low Mach number scheme for the Euler equations of gas dynamics. SIAM J. Sci. Comput. 36(6), B989–B1024 (2014)

  75. Notay, Y.: An aggregation-based algebraic multigrid method. Electron. Trans. Numer. Anal. 37, 123–146 (2010)

    MathSciNet  Google Scholar 

  76. Ogura, Y., Phillips, N.A.: Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci. 19(2), 173–179 (1962)

    Article  Google Scholar 

  77. Pareschi, L., Russo, G.: Implicit-explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25(112), 129–155 (2005)

    MathSciNet  Google Scholar 

  78. Powell, K.G.: An approximate Riemann solver for magnetohydrodynamics. In: Upwind and High-Resolution Schemes, pp. 570–583. Springer, Berlin (1997)

    Chapter  Google Scholar 

  79. Powell, K.G., Roe, P.L., Myong, R., Gombosi, T.: An upwind scheme for magnetohydrodynamics. In: 12th Computational Fluid Dynamics Conference, p. 1704 (1995)

  80. Rosenthal, C., Bogdan, T., Carlsson, M., Dorch, S., Hansteen, V., McIntosh, S., McMurry, A., Nordlund, Å., Stein, R.: Waves in the magnetized solar atmosphere. I. Basic processes and internetwork oscillations. Astrophys. J. 564(1), 508 (2002)

    Article  Google Scholar 

  81. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, pp. 325–432 (1998)

  82. Shun’ichi, G.: Singular limit of the incompressible ideal magneto-fluid motion with respect to the Alfvén number. Hokkaido Math. J. 19, 175–187 (1990)

    MathSciNet  Google Scholar 

  83. Stone, J.M., Gardiner, T.A., Teuben, P., Hawley, J.F., Simon, J.B.: Athena: a new code for astrophysical MHD. Astrophys. J. Suppl. Ser. 178(1), 137 (2008)

    Article  Google Scholar 

  84. Tang, M.: Second order method for isentropic Euler equation in the low Mach number regime. Kinet. Relat. Models 5(1), 155–184 (2012)

    Article  MathSciNet  Google Scholar 

  85. Tang, H., Xu, K.: A high-order gas-kinetic method for multidimensional ideal magnetohydrodynamics. J. Comput. Phys. 165(1), 69–88 (2000)

    Article  MathSciNet  Google Scholar 

  86. Tang, Q., Chacon, L., Kolev, T.V., Shadid, J.N., Tang, X.-Z.: An adaptive scalable fully implicit algorithm based on stabilized finite element for reduced visco-resistive MHD. J. Comput. Phys. 454, 110967 (2022)

    Article  MathSciNet  Google Scholar 

  87. Tavelli, M., Dumbser, M.: A pressure-based semi-implicit space-time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier–Stokes equations at all Mach numbers. J. Comput. Phys. 341, 341–376 (2017)

    Article  MathSciNet  Google Scholar 

  88. Thomann, A., Puppo, G., Klingenberg, C.: An all speed second order well-balanced IMEX relaxation scheme for the Euler equations with gravity. J. Comput. Phys. 420, 109723 (2020)

    Article  MathSciNet  Google Scholar 

  89. Vater, S., Klein, R.: A semi-implicit multiscale scheme for shallow water flows at low Froude number. Commun. Appl. Math. Comput. Sci. 13(2), 303–336 (2018)

    Article  MathSciNet  Google Scholar 

  90. Wu, K., Shu, C.-W.: A provably positive discontinuous Galerkin method for multidimensional ideal magnetohydrodynamics. SIAM J. Sci. Comput. 40(5), B1302–B1329 (2018)

    Article  MathSciNet  Google Scholar 

  91. Wu, K., Shu, C.-W.: Provably positive high-order schemes for ideal magnetohydrodynamics: analysis on general meshes. Numer. Math. 142(4), 995–1047 (2019)

    Article  MathSciNet  Google Scholar 

  92. Wu, K., Xing, Y.: Uniformly high-order structure-preserving discontinuous Galerkin methods for Euler equations with gravitation: positivity and well-balancedness. SIAM J. Sci. Comput. 43(1), A472–A510 (2021)

    Article  MathSciNet  Google Scholar 

  93. Wu, K., Jiang, H., Shu, C.-W.: Provably positive central discontinuous Galerkin schemes via geometric quasilinearization for ideal MHD equations. SIAM J. Numer. Anal. 61(1), 250–285 (2023)

    Article  MathSciNet  Google Scholar 

  94. Xing, Y., Shu, C.-W.: High order well-balanced WENO scheme for the gas dynamics equations under gravitational fields. SIAM J. Sci. Comput. 54(2), 645–662 (2013)

    Article  MathSciNet  Google Scholar 

  95. Xu, Z., Balsara, D.S., Du, H.: Divergence-free WENO reconstruction-based finite volume scheme for solving ideal MHD equations on triangular meshes. Commun. Comput. Phys. 19(4), 841–880 (2016)

    Article  MathSciNet  Google Scholar 

  96. Zank, G.P., Matthaeus, W.: Nearly incompressible fluids. II: Magnetohydrodynamics, turbulence, and waves. Phys. Fluids A Fluid Dyn. 5(1), 257–273 (1993)

    Article  MathSciNet  Google Scholar 

Download references

Funding

The first and the third authors are partially supported by National Key R & D Program of China No. 2022YFA1004500, NSFC grant No. 11971025, NSF of Fujian Province No. 2023J02003, and the Strategic Priority Research Program of Chinese Academy of Sciences Grant No. XDA25010401. The work of Kailiang Wu is partially supported by National Natural Science Foundation of China (Grant No. 12171227) and Shenzhen Science and Technology Program (Grant No. RCJC20221008092757098).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Wei Chen

  2. Department of Mathematics & SUSTech International Center for Mathematics, Southern University of Science and Technology, National Center for Applied Mathematics Shenzhen (NCAMS), Shenzhen, 518055, Guangdong, China

    Kailiang Wu

  3. School of Mathematical Sciences, Xiamen University, Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen, 361005, Fujian, People’s Republic of China

    Tao Xiong

Authors
  1. Wei Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kailiang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tao Xiong
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Wei Chen: Conceptualization, Formal analysis, Investigation, Methodology, Writing. Kailiang Wu: Conceptualization, Formal analysis, Methodology, Writing. Tao Xiong: Conceptualization, Formal analysis, Methodology, Writing, Funding acquisition.

Corresponding author

Correspondence to Tao Xiong.

Ethics declarations

Conflict of interest

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, W., Wu, K. & Xiong, T. High Order Structure-Preserving Finite Difference WENO Schemes for MHD Equations with Gravitation in all Sonic Mach Numbers. J Sci Comput 99, 36 (2024). https://doi.org/10.1007/s10915-024-02492-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-024-02492-7

Keywords

Search

Navigation