Rony Touma
Title :
Unstaggered Central schemes for Ideal and Shallow Water Magnetohydrodynamics
Abstract :
We propose one and two-dimensional unstaggered central finite volume methods for
solving hyperbolic systems of conservation laws. Based on Nessyahu and Tadmor's onedimensional central scheme, we construct a new class of unstaggered, second-order, nonoscillatory, central finite volume schemes. In contrast with the original (NT) central scheme
that evolves the numerical solution on an original grid (at even time steps) and a dual
staggered grid (at odd time steps), the proposed unstaggered central scheme (UCS) evolves
the numerical solution on a single grid and uses a "ghost" staggered grid to avoid the time
consuming resolution of the Riemann problems arising at the cell interfaces.
We apply the UCS method and solve some classical Ideal and shallow water
magnetohydrodynamic problems. To satisfy the solenoidal property of the magnetic
field/flux in the numerical solution, we use Evans and Hawley's constrained transport (CT)
method. The CT method applies easily with the UCS method; it deals directly with the
computed numerical solution, and preserves the second-order accuracy of the base
scheme. Even without the application of the CT procedure, the numerical base scheme does
not break down, and may even in some cases deliver reasonable results. However, to
ensure an admissible physical solution the CT treatment must be applied. The numerical
results show the efficiency and the potential of the proposed method and compare very
well with their corresponding ones appearing in the recent literature.
References :
C. R. Evans and J. F. Hawley, Simulation of magnetohydrodynamic flows: A constrained
transport method, Astrophys. J., 332, 659, 1988.
H.Nessyahu and E.Tadmor, Non-oscillatory central differencing for hyperbolic conservation
laws, J. Comp. Phys., 87, No. 2, pp. 408-463, 1990.
G.Jiang and E.Tadmor, Non- oscillatory central schemes for multidimensional hyperbolic
conservation laws, SIAM J. on Sc. Comp., 19, pp. 1892-1917, 1998.
P.Gilman Magnetohydrodynamic “Shallow Water” Equations for the Solar Tacholine, Astro.
J., 544, L79–L82, 2000.
R.Touma and P.Arminjon, Central finite volume schemes with constrained transport
divergence treatment for three-dimensional ideal MHD, J. Comp. Phys., 212, 617-636, 2006.