Block Incomplete LU-preconditioners for Implicit Solution of Advection Dominated Problems

In many astrophysical hydrodynamical and magnetohydro-dynamical problems explicit time integration is ineecient due to the dis-parate time scales, or because we are looking for a steady-state. Implicit schemes can ooer a viable alternative, which requires eecient solution of large linear systems. In multi-dimensional ows these linear systems cannot be solved with a direct method, so that iterative methods must be used. Since the physical problems are often advection dominated, the matrix is approximately anti-symmetric and non-diagonally dominant, thus preconditioning is necessary. We describe a new approach, in which a GMRES type iterative method is combined with an Eisenstat implementation of a relaxed form of a modiied Block Incomplete LU-decomposition as preconditioner. Paral-lelisation aspects of the algorithm are also discussed.