difficult part of creating a finite-volume scheme for MHD is the determination of the hyperbolic fluxes at each cell A new implicit algorithm is developed for solving the time-dependent, nonideal magnetohydrodynamic equations. It can also be used interface. Approximate Riemann solvers are a class of as an efficient relaxation scheme for steady state solutions. The methods for evaluating these fluxes that take into account algorithm is a finite-volume scheme that uses an approximate Riethe wave nature of hyperbolic equations. In the past, apmann solver for the hyperbolic fluxes and central differencing approximate Riemann solvers have been used extensively in plied on nested control volumes for the parabolic fluxes that arise the solution of the Euler and Navier-Stokes equations in from the non-ideal terms (i.e., resistivity and viscosity). In one dimension the scheme is second-order accurate in space and time. fluid dynamics. Recently, a number of explicit schemes In two or three dimensions, the accuracy is between first and second built around some type of approximate Riemann solver order. For the class of problems considered, the implicit formulation have been developed for the one-dimensional and multidiis stable for any size time step, thus allowing efficient tracking of mensional MHD equations [6].
slower transients. The implicit operator is inverted using a lower-While the MHD equations can be used to study plasma upper symmetric Gauss-Seidel iteration. Results from several test cases are presented that show good agreement with analytical solu-phenomena occurring on time scales as short as the transit tions and illustrate the advantages of the scheme. แฎ 1997 Academic time of a fast MHD wave, for many problems the important Press physics occurs on time scales that are much longer. For example, it can be shown that resistive tearing modes, which are important in studying fusion plasmas, evolve on * Alternate address is High Energy Plasma Division, Phillips Laboratory, Kirtland Air Force Base, New Mexico 87117-5776.
the full implicit operators became impractical, so various 231