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Note that in computing the above-mentioned Jacobian, the row and column corresponding to the normal component of the magnetic ﬁeld are dropped. The solution to the Riemann problem is then used to compute the ﬂuxes through the cell face. This method, as described above, can lead to numerical problems, that is, positivity is not guaranteed and negative pressures or densities may develop. If this occurs, we use the more robust and generally positivity preserving HLL ﬂux as given below. 10) where F is notationally a generic ﬂux through a cell face, UL and UR are the conserved quantities on the left/right side of the cell face, and λmin and λmax are, respectively, the minimum and maximum eigenvalues evaluated at the arithmetic average of the left and right states.
219:144–162, 2006. References 27 24. R. Samtaney, P. J. F. C. Jardin. An adaptive mesh semi-implicit conservative unsplit method for resistive MHD. In A. , editors, SciDAC 2005. San Francsico, June 26–30, 2005. 25. R. C. Jardin, P. F. Martin. 3D adaptive mesh reﬁnement simulations of pellet injection in tokamaks. Comput. Phys. , 164:220–228, 2004. 26. R. Strauss and W. Park. Pellet driven disruptions in tokamaks. Phys. Plasmas, 7:250–257, 2000. 27. G. Toth. The ∇ · B = 0 constraint in shock-capturing magnetohydrodynamics codes.
Traditional nondissipative ﬁnite difference schemes have large dispersive errors. These large spatial gradients are effectively treated with high-resolution upwind methods, which have been effectively used in shock capturing in computational ﬂuid dynamics. 2 MATHEMATICAL MODELS, EQUATIONS, AND NUMERICAL METHOD We begin by writing the equations of compressible resistive MHD in nearconservation form in cylindrical coordinates. These equations describe the conservation of mass, momentum, and energy, coupled with Maxwell’s equations for the evolution of the magnetic ﬁeld.