A solution accurate, efficient and stable unsplit staggered mesh scheme for three dimensional magnetohydrodynamics

Journal of Computational Physics

Volumn 243, Issue , 2013, Pages 269-292

  Lee, Dongwook ^a  

^a University of Chicago   (United States)

Author keywords

Constrained transport; Corner transport upwind; High order Godunov method; Large CFL number; Magnetohydrodynamics; MHD; Staggered mesh; Unsplit scheme

Indexed keywords

COST EFFECTIVENESS; ELECTRIC FIELDS; EVOLUTIONARY ALGORITHMS; IMAGE RECONSTRUCTION; MESH GENERATION; PHYSICAL PROPERTIES;

CFL NUMBER; CONSTRAINED TRANSPORTS; CORNER TRANSPORT UPWIND; GODUNOV METHOD; HIGH-ORDER; HIGH-ORDER GODUNOV METHOD; HIGHER-ORDER; LARGE CFL NUMBER; STAGGERED MESH; UNSPLIT SCHEME;

MAGNETOHYDRODYNAMICS;

EID: 84876332651     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2013.02.049     Document Type: Article

References (67)

1. Abgrall R. Approximation of the multidimensional Riemann problem of compressible fluid mechanics by a Roe type method. C.R. Acad. Sci. Paris Sér I Math. 1994, 319:499.
2. Almgren A.S., Bell J.B., Rendleman C.A., Zingale M. Low Mach number modeling of Type Ia supernovae, I. Hydrodynamics. Astrophys. J. 2006, 637:922-936.
3. Almgren A.S., Bell J.B., Rendleman C.A., Zingale M. CASTRO: A new compressible astrophysical solver I. Hydrodynamics and self-gravity. Astrophys. J. 2010, 715:1221-1238.
4. Balsara D.S., Spicer D.S. A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamics simulation. J. Comput. Phys. 1999, 149:270-292.
5. Balsara D.S., Shu C.-W. Monotonicity preserving weighted non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 2000, 160:405-452.
6. Balsara D.S. Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction. Astrophys. J. Suppl. 2004, 151:149-184.
7. Balsara D.S., Altmann C., Munz C.D., Dumbser M. A sub-cell based indicator for troubled zones in RKDG schemes and a novel class oh hybrid RKDG+HWENO schemes. J. Comput. Phys. 2007, 226:586-620.
8. Balsara D.S., Rumpf T., Dumbser M., Munz C.D. Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics. J. Comput. Phys. 2009, 228:2480-2516.
9. Balsara D.S. Multidimensional HLLE Riemann solver: application to Euler and magnetohydrodynamic flows. J. Comput. Phys. 2010, 229:1970-1993.
10. Balsara D.S. A two-dimensional HLLC Riemann solver for conservation laws: application to Euler and magnetohydrodynamic flows. J. Comput. Phys. 2012, 231:7476-7503.
11. Balsara D.S., Meyer C., Dumbser M., Du H., Xu Z. Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes - Speed comparisons with Runge-Kutta methods. J. Comput. Phys. 2013, 235:934-969.
12. T.J. Barth, P.O. Frederickson, Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA Paper no. 90-0013, 1990.
13. Bell J., Dawson C.N., Shubin G.R. An unsplit, higher order Godunov method for scalar conservation laws in multiple dimensions. J. Comput. Phys. 1988, 74:1-24.
14. Brio M., Zakharian A.R., Webb G.M. Two-dimensional Riemann solver for Euler equations of gas dynamics. J. Comput. Phys. 2001, 167:177-195.
15. Colella P. Multidimensional upwind methods for hyperbolic conservation laws. J. Comput. Phys. 1990, 87:171-200.
16. Colella P., Woodward P. The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys. 1984, 54:174-201.
17. Crockett R.K., Colella P., Fisher R.T., Klein R.I., McKee C.F. An unsplit, cell-centered Godunov method for ideal MHD. J. Comput. Phys. 2005, 203:422-448.
18. Dai W., Woodward P. A simple finite difference scheme for multidimensional magnetohydrodynamics. J. Comput. Phys. 1998, 142:331-369.
19. Dubey A., Antypas K., Ganapathy M.K., Reid L.B., Riley K.M., Sheeler D., Siegel A., Weide K. Extensible component based architecture for FLASH, a massively parallel, multiphysics simulation code. Parallel Comput. 2009, 35(10-11):512-522.
20. Dumbser M., Käser M. Arbitary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J. Comput. Phys. 2007, 221:693-723.
21. Dumbser M., Käser M., Titarev V.A., Toro E.F. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J. Comput. Phys. 2007, 226:204-243.
22. Dumbser M., Balsara D., Toro E.F., Munz C.D. A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys. 2008, 227:8209-8253.
23. Dumbser M., Enaux C., Toro E.F. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J. Comput. Phys. 2008, 227:3971-4001.
24. Evans C.R., Hawley J.F. Simulation of magnetohydrodynamic flows a constrained transport method. Astrophys. J. 1988, 332:659-677.
25. Fey M. Multidimensional upwinding 1. The method of transport for solving the Euler equations. J. Comput. Phys. 1998, 143:159-180.
26. Fey M. Multidimensional upwinding 2. Decomposition of the Euler equation into advection equation. J. Comput. Phys. 1998, 143:181-199.
27. FLASH. , 2013. http://flash.uchicago.edu.
28. Gardiner T., Stone J. An unsplit Godunov method for ideal MHD via constrained transport. J. Comput. Phys. 2005, 205:509-539.
29. Gardiner T., Stone J. An unsplit Godunov method for ideal MHD via constrained transport in three dimensions. J. Comput. Phys. 2008, 227:4123-4141.
30. Gilquin H., Laurens J., Rosier C. Multidimensional Riemann problems for linear hyperbolic systems. Notes Numer. Fluid Mech. 1993, 43:284.
31. Hanawa T., Mikami H., Matsmoto T. Improving shock irregularities based on the characteristic of MHD equations. J. Comput. Phys. 2008, 227:7952-7976.
32. Harten A., Osher S., Engquist B., Charkaravarthy S. Some results on uniformly high-order accurate essentially nonoscillatory schemes. Appl. Numer. Math. 1987, 2:347-377.
33. Harten A., Engquist B., Osher S., Charkaravarthy S. Uniformly high-order essentially non-oscillatory schemes III. J. Comput. Phys. 1987, 71:231-303.
34. Helzel C., Rossmanith J.A., Taetz B. An unstaggered constrained transport method for the 3D ideal magnetohydrodynamics equations. J. Comput. Phys. 2011, 230:3803-3829.
35. Hu C., Shu C.W. Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 1999, 150:97-127.
36. Jiang G., Shu C.W. Efficient implementation of weighted ENO schemes. J. Comput. Phys. 1996, 126:202-228.
37. Kraichnan R.H. Inertial ranges in two-dimensional turbulence. Phys. Fluids 1967, 10:1417-1423.
38. Lee D., Deane A.E. An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics. J. Comput. Phys. 2009, 228:952-975.
39. D. Lee, A. Dubey, K. Olson, K. Weide, K. Antypas, Exploiting the extensibility of the FLASH code architecture for unsplit time integration, Numerical modeling of space plasma flows: Astronum-2009, in: APS Conference Series, vol. 429, 2010, pp. 247-252.
40. R.J. LeVeque, Numerical methods for conservation laws, Birkhäuser, 1992.
41. S. Li, Note on upwinding constrained transport method for ideal magnetohydrodynamics, Los Alamos, Report, LA-UR-03-8925, 2003.
42. Liu X.-D., Osher S., Chan T. Weighted essentially non-oscillatory schemes. J. Comput. Phys. 1994, 115:200-212.
43. Londrillo P., Del Zanna L. On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method. J. Comput. Phys. 2004, 195:14-48.
44. Mignone A., Tzeferacos P. A second-order unsplit Godunov scheme for cell-centered MHD the CTU-GLM scheme. J. Comput. Phys. 2010, 229:2117-2138.
45. Miniati F., Martin D.F. Constrained-transport magnetohydrodynamics with adaptive mesh refinement in CHARM. Astrophys. J. Suppl. 2011, 195:5-25.
46. Miyoshi T., Kusano K. A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics. J. Comput. Phys. 2005, 208:315-344.
47. Orszag S.A., Tang C.M. Small-scale structure of two-dimensional magnetohydrodynamics turbulence. J. Fluid Mech. 1979, 90:129-143.
48. Pandolfi M., D'Ambrosio D. Numerical instabilities in upwind methods: analysis and cures for the Carbuncle phenomenon. J. Comput. Phys. 2001, 166:271-301.
49. K. G. Powell, A Riemann solver for ideal MHD: that works in more than one dimension, ICASE Report, 94-24, Langely, VA, 1994.
50. Quirk J. A contribution to the great Riemann solver debate. Int. J. Numer. Methods Fluids 1994, 18:555-574.
51. Roe P.L. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 1981, 43:357-372.
52. Rumsey C.L., van Leer B., Roe P.L. A multidimensional flux function with applications to the Euler and Navier-Stokes equations. J. Comput. Phys. 1993, 105:306-323.
53. Saltzman J. An unsplit 3D upwind method for hyperbolic conservation laws. J. Comput. Phys. 1994, 115:153-168.
54. Sanders R., Morano E., Druguet M. Multidimensional dissipation for upwind schemes: stability and applications to gas dynamics. J. Comput. Phys. 1998, 145:511-537.
55. Shu C.-W., Osher S.J. Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 1988, 77:439-471.
56. Shu C.-W., Osher S.J. Efficient implementation of essentially non-oscillatory shock capturing schemes II. J. Comput. Phys. 1989, 83:32-78.
57. Stone J., Gardiner T., Teuben P., Hawley J., Simon J. Athena: a new code for astrophysical MHD. Apstrophy. J. Suppl. 2008, 178:137-177.
58. Suresh A., Huynh H.T. Accurate monotonicity preserving scheme with Runge-Kutta time-stepping. J. Comput. Phys. 1997, 136:83-99.
59. Titarev V.A., Toro E.F. ADER: arbitrary high order Godunov approach. J. Sci. Comput. 2002, 17(1-4):609-618.
60. Titarev V.A., Toro E.F. ADER schemes for three-dimensional nonlinear hyperbolic systems. J. Comput. Phys. 2005, 204:715-736.
61. Toro E.F. Riemann solvers and numerical methods for fluid dynamics, a practical introduction 2009, Springer. Third ed.
62. Toro E.F., Titarev V.A. Derivative Riemann solvers for systems of conservation laws and ADER methods. J. Comput. Phys. 2006, 212(1):150-165.
63. Tóth G. The ∇; · B = 0 constraint in shock-capturing magnetohydrodynamics codes. J. Comput. Phys. 2000, 161:605-656.
64. van Leer B. Upwind and high-resolution methods for compressible flow: from donor cell to residual-distribution schemes. Commun. Comput. Phys. 2006, 1:192-206.
65. Wendroff B. A two-dimensional HLLE Riemann solver and associated Godunov-type difference scheme for gas dynamics. Comput. Math. Appl. 1999, 38:175-185.
66. Zachary A., Malagoli A., Colella P. A higher-order Godunov method for multidimensional ideal magnetohydrodynamics. J. Comput. Phys. 1994, 2:263-284.
67. Zhang Y.-T., Shu C.-W. High order WENO schemes for Hamilton-Jacobi equations on triangular meshes. SIAM J. Sci. Comput. 2003, 24:1005-1030.