Block incomplete LU-preconditioners for implicit solution of advection dominated problems

In many astrophysical hydrodynamical and magnetohydrodynamical problems explicit time integration is inefficient due to the disparate time scales, or because we are looking for a steady-state. Implicit schemes can offer a viable alternative, which requires efficient solution of large linear systems. In multi-dimensional flows 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 modified Block Incomplete LU-decomposition as preconditioner. Parallelisation aspects of the algorithm are also discussed.