An Implicit Scheme for Nonideal Magnetohydrodynamics

A new implicit algorithm is developed for solving the time-dependent, nonideal magnetohydrodynamic equations. It can also be used as an efficient relaxation scheme for steady state solutions. The algorithm is a finite-volume scheme that uses an approximate Riemann solver for the hyperbolic fluxes and central differencing applied on nested control volumes for the parabolic fluxes that arise from the non-ideal terms (i.e., resistivity and viscosity). In one dimension the scheme is second-order accurate in space and time. In two or three dimensions, the accuracy is between first and second order. For the class of problems considered, the implicit formulation is stable for any size time step, thus allowing efficient tracking of slower transients. The implicit operator is inverted using a lower–upper symmetric Gauss–Seidel iteration. Results from several test cases are presented that show good agreement with analytical solutions and illustrate the advantages of the scheme.