Constraint preserving schemes using potential-based fluxes. III. Genuinely multi-dimensional schemes for MHD equations∗∗∗

Siddhartha Mishra; Eitan Tadmor

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 3, page 661-680
  • ISSN: 0764-583X

Abstract

top
We design efficient numerical schemes for approximating the MHD equations in multi-dimensions. Numerical approximations must be able to deal with the complex wave structure of the MHD equations and the divergence constraint. We propose schemes based on the genuinely multi-dimensional (GMD) framework of [S. Mishra and E. Tadmor, Commun. Comput. Phys. 9 (2010) 688–710; S. Mishra and E. Tadmor, SIAM J. Numer. Anal. 49 (2011) 1023–1045]. The schemes are formulated in terms of vertex-centered potentials. A suitable choice of the potential results in GMD schemes that preserve a discrete version of divergence. First- and second-order divergence preserving GMD schemes are tested on a series of benchmark numerical experiments. They demonstrate the computational efficiency and robustness of the GMD schemes.

How to cite

top

Mishra, Siddhartha, and Tadmor, Eitan. "Constraint preserving schemes using potential-based fluxes. III. Genuinely multi-dimensional schemes for MHD equations∗∗∗." ESAIM: Mathematical Modelling and Numerical Analysis 46.3 (2012): 661-680. <http://eudml.org/doc/276382>.

@article{Mishra2012,
abstract = {We design efficient numerical schemes for approximating the MHD equations in multi-dimensions. Numerical approximations must be able to deal with the complex wave structure of the MHD equations and the divergence constraint. We propose schemes based on the genuinely multi-dimensional (GMD) framework of [S. Mishra and E. Tadmor, Commun. Comput. Phys. 9 (2010) 688–710; S. Mishra and E. Tadmor, SIAM J. Numer. Anal. 49 (2011) 1023–1045]. The schemes are formulated in terms of vertex-centered potentials. A suitable choice of the potential results in GMD schemes that preserve a discrete version of divergence. First- and second-order divergence preserving GMD schemes are tested on a series of benchmark numerical experiments. They demonstrate the computational efficiency and robustness of the GMD schemes.},
author = {Mishra, Siddhartha, Tadmor, Eitan},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Multidimensional evolution equations; magnetohydrodynamics; constraint transport; central difference schemes; potential-based fluxes; multidimensional evolution equations},
language = {eng},
month = {1},
number = {3},
pages = {661-680},
publisher = {EDP Sciences},
title = {Constraint preserving schemes using potential-based fluxes. III. Genuinely multi-dimensional schemes for MHD equations∗∗∗},
url = {http://eudml.org/doc/276382},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Mishra, Siddhartha
AU - Tadmor, Eitan
TI - Constraint preserving schemes using potential-based fluxes. III. Genuinely multi-dimensional schemes for MHD equations∗∗∗
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/1//
PB - EDP Sciences
VL - 46
IS - 3
SP - 661
EP - 680
AB - We design efficient numerical schemes for approximating the MHD equations in multi-dimensions. Numerical approximations must be able to deal with the complex wave structure of the MHD equations and the divergence constraint. We propose schemes based on the genuinely multi-dimensional (GMD) framework of [S. Mishra and E. Tadmor, Commun. Comput. Phys. 9 (2010) 688–710; S. Mishra and E. Tadmor, SIAM J. Numer. Anal. 49 (2011) 1023–1045]. The schemes are formulated in terms of vertex-centered potentials. A suitable choice of the potential results in GMD schemes that preserve a discrete version of divergence. First- and second-order divergence preserving GMD schemes are tested on a series of benchmark numerical experiments. They demonstrate the computational efficiency and robustness of the GMD schemes.
LA - eng
KW - Multidimensional evolution equations; magnetohydrodynamics; constraint transport; central difference schemes; potential-based fluxes; multidimensional evolution equations
UR - http://eudml.org/doc/276382
ER -

References

top
  1. R. Artebrant and M. Torrilhon, Increasing the accuracy of local divergence preserving schemes for MHD. J. Comput. Phys.227 (2008) 3405–3427.  
  2. J. Bálbas and E. Tadmor, Non-oscillatory central schemes for one and two-dimensional magnetohydrodynamics II : High-order semi-discrete schemes. SIAM. J. Sci. Comput.28 (2006) 533–560.  
  3. J. Bálbas, E. Tadmor and C.C. Wu, Non-oscillatory central schemes for one and two-dimensional magnetohydrodynamics I. J. Comput. Phys.201 (2004) 261–285.  
  4. D.S. Balsara, Divergence free adaptive mesh refinement for magnetohydrodynamics. J. Comput. Phys.174 (2001) 614–648.  
  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) 270–292.  
  6. J.B. Bell, P. Colella and H.M. Glaz, A second-order projection method for the incompressible Navier-Stokes equations. J. Comput. Phys.85 (1989) 257–283.  
  7. F. Bouchut, C. Klingenberg and K. Waagan, A multi-wave HLL approximate Riemann solver for ideal MHD based on relaxation I- theoretical framework. Numer. Math.108 (2007) 7–42.  
  8. J.U. Brackbill and D.C. Barnes, The effect of nonzero DivB on the numerical solution of the magnetohydrodynamic equations. J. Comput. Phys.35 (1980) 426–430.  
  9. M. Brio and C.C. Wu, An upwind differencing scheme for the equations of ideal MHD. J. Comput. Phys.75 (1988) 400–422.  
  10. A.J. Chorin, Numerical solutions of the Navier-Stokes equations. Math. Comput.22 (1968) 745–762.  
  11. W. Dai and P.R. Woodward, A simple finite difference scheme for multi-dimensional magnetohydrodynamic equations. J. Comput. Phys.142 (1998) 331–369.  
  12. H. Deconnik, P.L. Roe and R. Struijs, A multi-dimensional generalization of Roe’s flux difference splitter for Euler equations. Comput. Fluids22 (1993) 215.  
  13. 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) 645–673.  
  14. C. Evans and J.F. Hawley, Simulation of magnetohydrodynamic flow : a constrained transport method. Astrophys. J.332 (1998) 659.  
  15. M. Fey, Multi-dimensional upwingding. (I) The method of transport for solving the Euler equations. J. Comput. Phys.143 (1998) 159–180.  
  16. M. Fey, Multi-dimensional upwingding.(II) Decomposition of Euler equations into advection equations. J. Comput. Phys.143 (1998) 181–199.  
  17. F. Fuchs, S. Mishra and N.H. Risebro, Splitting based finite volume schemes for ideal MHD equations. J. Comput. Phys.228 (2009) 641–660.  
  18. F. Fuchs, A. McMurry, S. Mishra, N.H. Risebro and K. Waagan, Finite volume methods for wave propagation in stratified magneto-atmospheres. Commun. Comput. Phys.7 (2010) 473–509.  
  19. F. Fuchs, A.D. McMurry, S. Mishra, N.H. Risebro and K. Waagan, Approximate Riemann solver and robust high-order finite volume schemes for the MHD equations in multi-dimensions. Commun. Comput. Phys.9 (2011) 324–362.  
  20. S. Gottlieb, C.W. Shu and E. Tadmor, High order time discretizations with strong stability property. SIAM. Rev.43 (2001) 89–112.  
  21. K.F. Gurski, An HLLC-type approximate Riemann solver for ideal Magneto-hydro dynamics. SIAM. J. Sci. Comput.25 (2004) 2165–2187.  
  22. A. Harten, B. Engquist, S. Osher and S.R. Chakravarty, Uniformly high order accurate essentially non-oscillatory schemes. J. Comput. Phys.71 (1987) 231–303.  
  23. A. Kurganov and E. Tadmor, New high resolution central schemes for non-linear conservation laws and convection-diffusion equations. J. Comput. Phys.160 (2000) 241–282.  
  24. R.J. LeVeque, Wave propagation algorithms for multi-dimensional hyperbolic systems, J. Comput. Phys.131 (1997) 327–353.  
  25. R.J. LeVeque, Finite volume methods for hyperbolic problems. Cambridge university press, Cambridge (2002).  
  26. T.J. Linde, A three adaptive multi fluid MHD model for the heliosphere. Ph.D. thesis, University of Michigan, Ann-Arbor (1998).  
  27. M. Lukacova-Medvidova, K.W. Morton and G. Warnecke, Evolution Galerkin methods for Hyperbolic systems in two space dimensions. Math. Comput.69 (2000) 1355–1384.  
  28. M. Lukacova-Medvidova, J. Saibertova and G. Warnecke, Finite volume evolution Galerkin methods for Non-linear hyperbolic systems. J. Comput. Phys.183 (2003) 533–562.  
  29. S. Mishra and E. Tadmor, Constraint preserving schemes using potential-based fluxes. I. Multi-dimensional transport equations. Commun. Comput. Phys.9 (2010) 688–710.  
  30. S. Mishra and E. Tadmor, Constraint preserving schemes using potential-based fluxes. II. Genuinely multi-dimensional systems of conservation laws. SIAM J. Numer. Anal.49 (2011) 1023–1045.  
  31. A. Mignoneet al., Pluto : A numerical code for computational astrophysics. Astrophys. J. Suppl.170 (2007) 228–242.  
  32. T. Miyoshi and K. Kusano, A multi-state HLL approximate Riemann solver for ideal magneto hydro dynamics. J. Comput. Phys.208 (2005) 315–344.  
  33. H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys.87 (1990) 408–463.  
  34. S. Noelle, The MOT-ICE : A new high-resolution wave propagation algorithm for multi-dimensional systems of conservation laws based on Fey’s method of transport. J. Comput. Phys.164 (2000) 283–334.  
  35. K.G. Powell, An approximate Riemann solver for magneto-hydro dynamics (that works in more than one space dimension). Technical report, ICASE, Langley, VA (1994) 94–24.  
  36. K.G. Powell, P.L. Roe, T.J. Linde, T.I. Gombosi and D.L. De zeeuw, A solution adaptive upwind scheme for ideal MHD. J. Comput. Phys.154 (1999) 284–309.  
  37. P.L. Roe and D.S. Balsara, Notes on the eigensystem of magnetohydrodynamics. SIAM. J. Appl. Math.56 (1996) 57–67.  
  38. J. Rossmanith, A wave propagation method with constrained transport for shallow water and ideal magnetohydrodynamics. Ph.D. thesis, University of Washington, Seattle (2002).  
  39. D.S. Ryu, F. Miniati, T.W. Jones and A. Frank, A divergence free upwind code for multidimensional magnetohydrodynamic flows. Astrophys. J.509 (1998) 244–255.  
  40. C.W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory schemes – II. J. Comput. Phys.83 (1989) 32–78.  
  41. E. Tadmor, Approximate solutions of nonlinear conservation laws, in Advanced Numerical approximations of NonlinearHyperbolic equations, edited by A. Quarteroni. Lecture notes in Mathematics, Springer Verlag (1998) 1–149.  
  42. M. Torrilhon, Locally divergence preserving upwind finite volume schemes for magnetohyrodynamic equations. SIAM. J. Sci. Comput.26 (2005) 1166–1191.  
  43. M. Torrilhon and M. Fey, Constraint-preserving upwind methods for multidimensional advection equations. SIAM. J. Numer. Anal.42 (2004) 1694–1728.  
  44. G. Toth, The DivB = 0 constraint in shock capturing magnetohydrodynamics codes. J. Comput. Phys.161 (2000) 605–652.  
  45. B. van Leer, Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov’s method. J. Comput. Phys.32 (1979) 101–136.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.