Geodesic
Piecewise parabolic method on a local stencil for ideal magnetohydrodynamics
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 3, pp. 505-528 
M. V. Popov; S. D. Ustyugov. Piecewise parabolic method on a local stencil for ideal magnetohydrodynamics. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 3, pp. 505-528. http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_3_a10/
@article{ZVMMF_2008_48_3_a10,
     author = {M. V. Popov and S. D. Ustyugov},
     title = {Piecewise parabolic method on a~local stencil for ideal magnetohydrodynamics},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {505--528},
     year = {2008},
     volume = {48},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_3_a10/}
}
TY  - JOUR
AU  - M. V. Popov
AU  - S. D. Ustyugov
TI  - Piecewise parabolic method on a local stencil for ideal magnetohydrodynamics
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2008
SP  - 505
EP  - 528
VL  - 48
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_3_a10/
LA  - ru
ID  - ZVMMF_2008_48_3_a10
ER  - 
%0 Journal Article
%A M. V. Popov
%A S. D. Ustyugov
%T Piecewise parabolic method on a local stencil for ideal magnetohydrodynamics
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2008
%P 505-528
%V 48
%N 3
%U http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_3_a10/
%G ru
%F ZVMMF_2008_48_3_a10

Voir la notice de l'article provenant de la source Math-Net.Ru

A numerical scheme based on the piecewise parabolic method on a local stencil (PPML) is proposed for solving the ideal magnetohydrodynamic (MHD) equations. The method makes use of the conservation of Riemann invariants along the characteristics of the MHD equations. As a result, a local stencil can be used to construct a numerical solution. This approach improves the dissipative properties of the numerical scheme and is convenient in the case of adaptive grids. The basic stages in the design of the scheme are illustrated in the two-dimensional case. The conservation of the solenoidal property of the magnetic field is discussed. The scheme is tested using several typical MHD problems.

[1] Ustyugov S. D., Popov M. V., “Kusochno-parabolicheskii metod na lokalnom shablone dlya zadach gazovoi dinamiki”, Zh. vychisl. matem. i matem. fiz., 47:12 (2007), 2055–2075 | MR

[2] Collela P., Woodward P., “The piecewise parabolic method for gas-dynamical simulations”, J. Comput. Phys., 54 (1984), 174–201 | DOI

[3] Brio M., Wu C. C., “An upwind differencing scheme for the equations of ideal magnetohydrodynamics”, J. Comput. Phys., 75 (1988), 400–422 | DOI | MR | Zbl

[4] Powell K. G., Roe P. L., Linde T. J. et al., “A solution-adaptive upwind scheme for ideal magnetohydrodynamics”, J. Comput. Phys., 154 (1999), 284–309 | DOI | MR | Zbl

[5] Roe P. L., “Characteristic-based schemes for the Euler equations”, Ann. Rev. Fluid Mech., 18 (1986), 337–365 | DOI | MR | Zbl

[6] Tóth G., “The $\bigtriangledown\cdot\mathbf B=0$ constraint in shock-capturins magnetohydrodynamics codes”, J. Comput. Phys., 161 (2000), 605–652 | DOI | MR | Zbl

[7] Gombosi T. I., Powell K. G., De Zeeuw D. L., “Axisymmetric modeling of cometary mass loading on an adaptively refined grid: MHD results”, J. Geophys. Res., 99 (1994), 21525–21539 | DOI

[8] Brackbill J. U., Barnes D. C., “The effect of nonzero $\nabla\cdot\mathbf B$ on the numerical solution of the magnetohydrodynamic equations”, J. Comput. Phys., 35 (1980), 426–430 | DOI | MR | Zbl

[9] Evans C. R., Hawley J. F., “Simulation of magnetohydrodynamic flows: A constrained transport method”, Astrophys. J., 332 (1988), 659–677 | DOI

[10] 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 (1999), 270–292 | DOI | MR | Zbl

[11] Barth T. J., On unstructured grids and solvers, Comput. Fluid Dynamics. Lect. Ser., 1990-04, Von Kármán Institute for Fluid Dynamics, 1990

[12] Suresh A., Huynh H. T., “Accurate monotonicity preserving scheme with Runge–Kutta time-stepping”, J. Comput. Phys., 136 (1997), 83–99 | DOI | MR | Zbl

[13] Balsara D. S., Shu C.-W., “Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy”, J. Comput. Phys., 160 (2000), 405–452 | DOI | MR | Zbl

[14] Jiang G.-S., Wu C. C., “A high-order WENO finite difference scheme for the equation of ideal magnetohydrodynamics”, J. Comput. Phys., 150 (1999), 561–594 | DOI | MR | Zbl

[15] Balsara D. S., “Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction”, Astrophys. J. Suppl., 151 (2004), 149–184 | DOI

[16] Han J., Tang H., “An adaptive moving mesh method for two-dimensional ideal magnetohydrodynamics”, J. Comput. Phys., 220 (2007), 791–812 | DOI | MR | Zbl

[17] Orszag A., Tang C. M., “Small-scale structure of two-dimensional magnetohydrodynamic turbulence”, J. Fluid Mech, 90 (1979), 129–143 | DOI