Advertisement

Journal of Scientific Computing

, Volume 80, Issue 1, pp 692–716 | Cite as

High-Order Low-Dissipation Targeted ENO Schemes for Ideal Magnetohydrodynamics

  • Lin FuEmail author
  • Qi Tang
Technical Note

Abstract

The recently proposed targeted ENO (TENO) schemes (Fu et al. J Comput Phys 305:333–359, 2016) are demonstrated to feature the controllable low numerical dissipation and sharp shock-capturing property in compressible gas dynamic simulations. However, the application of low-dissipation TENO schemes to ideal magnetohydrodynamics (MHD) is not straightforward. The complex interaction between fluid mechanics and electromagnetism induces extra numerical challenges, including simultaneously preserving the ENO-property, maintaining good numerical robustness and low dissipation as well as controlling divergence errors. In this paper, based on an unstaggered constrained transport framework to control the divergence error, we extend a set of high-order low-dissipation TENO schemes ranging from 5-point to 8-point stencils to solving the ideal MHD equations. A unique set of built-in parameters for each TENO scheme is determined. Different from the TENO schemes in Fu et al.  (2016), a modified scale-separation formula is developed. The new formula can achieve stronger scale separation, and it is simpler and more efficient than the previous version as the computation cost of high-order global smoothness measure \({\tau _K}\) is avoided. The performances of tailored schemes are systematically studied by several benchmark simulations. Numerical experiments demonstrate that the TENO schemes in the constrained transport framework are promising to simulate more complex MHD flows.

Keywords

TENO WENO High-order accuracy Low dissipation MHD 

Notes

Acknowledgements

The first author is funded by U.S. Air Force Office of Scientific Research (AFOSR) (Grant No. 1194592-1-TAAHO). The second author is supported by the Eliza Ricketts Postdoctoral Fellowship.

References

  1. 1.
    Acker, F., Borges, R.D.R., Costa, B.: An improved WENO-Z scheme. J. Comput. Phys. 313, 726–753 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Arshed, G.M., Hoffmann, K.A.: Minimizing errors from linear and nonlinear weights of WENO scheme for broadband applications with shock waves. J. Comput. Phys. 246, 58–77 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Balsara, D.S.: Divergence-free adaptive mesh refinement for magnetohydrodynamics. J. Comput. Phys. 174(2), 614–648 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Balsara, D.S.: A two-dimensional HLLC Riemann solver for conservation laws: application to Euler and magnetohydrodynamic flows. J. Comput. Phys. 231(22), 7476–7503 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Balsara, D.S., Kim, J.: A comparison between divergence-cleaning and staggered-mesh formulations for numerical magnetohydrodynamics. Astrophys. J. 602(2), 1079 (2004)CrossRefGoogle Scholar
  6. 6.
    Brackbill, J.U., Barnes, D.: The effect of nonzero \(\nabla \cdot B\) on the numerical solution of the magnetohydrodynamic equation. J. Comput. Phys. 35(3), 426–430 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Brio, M., Wu, C.C.: An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 75(2), 400–422 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cheng, Y., Li, F., Qiu, J., Xu, L.: Positivity-preserving DG and central DG methods for ideal MHD equations. J. Comput. Phys. 238, 255–280 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Dahlburg, R., Picone, J.: Evolution of the Orszag–Tang vortex system in a compressible medium. I. Initial average subsonic flow. Phys. Fluids B Plasma Phys. (1989–1993) 1(11), 2153–2171 (1989)MathSciNetCrossRefGoogle Scholar
  14. 14.
    De Sterck, H.: Multi-dimensional upwind constrained transport on unstructured grids for ’shallow water’ magnetohydrodynamics. In: AIAA Computational Fluid Dynamics Conference, 15th, Anaheim, CA (2001)Google Scholar
  15. 15.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Dumbser, M., Balsara, D.S.: A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems. J. Comput. Phys. 304, 275–319 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Evans, C.R., Hawley, J.F.: Simulation of magnetohydrodynamic flows-A constrained transport method. Astrophys. J. 332, 659–677 (1988)CrossRefGoogle Scholar
  18. 18.
    Fey, M., Torrilhon, M.: A constrained transport upwind scheme for divergence-free advection. In: Hyperbolic Problems: Theory, Numerics, Applications, pp. 529–538. Springer, Berlin (2003)Google Scholar
  19. 19.
    Fu, L.: A low-dissipation finite-volume method based on a new TENO shock-capturing scheme. Comput. Phys. Commun. 235, 25–39 (2019)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Fu, L., Hu, X.Y., Adams, N.A.: A family of high-order targeted ENO schemes for compressible-fluid simulations. J. Comput. Phys. 305, 333–359 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Fu, L., Hu, X.Y., Adams, N.A.: Targeted ENO schemes with tailored resolution property for hyperbolic conservation laws. J. Comput. Phys. 349, 97–121 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Fu, L., Hu, X.Y., Adams, N.A.: A new class of adaptive high-order targeted ENO schemes for hyperbolic conservation laws. J. Comput. Phys. 374, 724–751 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Haimovich, O., Frankel, S.H.: Numerical simulations of compressible multicomponent and multiphase flow using a high-order targeted ENO (TENO) finite-volume method. Comput. Fluids 146, 105–116 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Jiang, G.-S., Peng, D.: Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21(6), 2126–2143 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Jiang, G.S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Kawai, S.: Divergence-free-preserving high-order schemes for magnetohydrodynamics: an artificial magnetic resistivity method. J. Comput. Phys. 251, 292–318 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Li, F., Shu, C.-W.: Locally divergence-free discontinuous Galerkin methods for MHD equations. J. Sci. Comput. 22(1–3), 413–442 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Li, F., Xu, L., 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)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Orszag, S.A., Tang, C.-M.: Small-scale structure of two-dimensional magnetohydrodynamic turbulence. J. Fluid Mech. 90(01), 129–143 (1979)CrossRefGoogle Scholar
  33. 33.
    Powell, K.G., Roe, P.L., Linde, T.J., Gombosi, T.I., De Zeeuw, D.L.: A solution-adaptive upwind scheme for ideal magnetohydrodynamics. J. Comput. Phys. 154(2), 284–309 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Ren, Y.-X., Liu, M., Zhang, H.: A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws. J. Comput. Phys. 192(2), 365–386 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Rossmanith, J.A.: An unstaggered, high-resolution constrained transport method for magnetohydrodynamic flows. SIAM J. Sci. Comput. 28(5), 1766–1797 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Ryu, D., Jones, T.: Numerical magetohydrodynamics in astrophysics: algorithm and tests for one-dimensional flow. Astrophys. J. 442, 228–258 (1995)CrossRefGoogle Scholar
  37. 37.
    Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J. Comput. Phys. 83, 32–78 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Sun, Z., Inaba, S., Xiao, F.: Boundary Variation Diminishing (BVD) reconstruction: a new approach to improve Godunov schemes. J. Comput. Phys. 322, 309–325 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Susanto, A.: High-Order Finite-Volume Schemes for Magnetohydrodynamics, Ph.D. thesis (2014)Google Scholar
  40. 40.
    Torrilhon, M.: Uniqueness conditions for Riemann problems of ideal magnetohydrodynamics. J. Plasma Phys. 69(03), 253–276 (2003)CrossRefGoogle Scholar
  41. 41.
    Tóth, G.: The \(\nabla \cdot B= 0\) constraint in shock-capturing magnetohydrodynamics codes. J. Comput. Phys. 161(2), 605–652 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Yang, Y., Wan, M., Shi, Y., Yang, K., Chen, S.: A hybrid scheme for compressible magnetohydrodynamic turbulence. J. Comput. Phys. 306, 73–91 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Yee, H.C., Sjögreen, B.: Non-linear filtering and limiting in high order methods for ideal and non-ideal MHD. J. Sci. Comput. 27(1–3), 507–521 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Yee, H.C., Sjögreen, B.: Adaptive filtering and limiting in compact high order methods for multiscale gas dynamics and MHD systems. Comput. Fluids 37(5), 593–619 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Center for Turbulence ResearchStanford UniversityStanfordUSA
  2. 2.Department of Mathematical SciencesRensselaer Polytechnic InstituteTroyUSA

Personalised recommendations