Implicit and Semi - Implicit Schemes :

We formulate general guidelines to extend an explicit code with a great variety of implicit and semi-implicit time integration schemes. The discussion is based on their speciic implementation in the Versatile Advection Code, which is a general purpose software package for solving systems of non-linear hyperbolic (and/or parabolic) partial diierential equations, using standard high-resolution shock-capturing schemes. For all combinations of explicit high-resolution schemes with implicit and semi-implicit treatments, we show how second order spatial and temporal accuracy for the smooth part of the solutions can be maintained. We discuss strategies to obtain steady state and time accurate solutions implicitly. The implicit and semi-implicit schemes require the solution of large linear systems containing the Jacobian matrix. The Jacobian matrix itself is calculated numerically to ensure the generality of our implementation. We discuss three options in terms of applicability, storage requirements, and computational eeciency. One option is the easily implemented matrix-free approach, but the Jacobian matrix can also be calculated by using a general grid masking algorithm, or by an ee-cient implementation for a speciic Lax-Friedrich type Total Variation Diminishing spatial discretization. The choice of the linear solver depends on the dimensionality of the problem. In one dimension a direct block tridiagonal solver can be applied, while in more than one spatial dimension a Conjugate Gradient-type iterative solver is used. For advection dominated problems , preconditioning is needed to accelerate the convergence of the iterative schemes. We implemented the Modiied Block Incomplete LU-preconditioner which performs very well. Examples from two-dimensional hydrodynamic and magnetohydrodynamic computations are given. They model transonic stellar outtow and recover the complex magnetohydrodynamic bow shock ow in the switch-on regime found in De Sterck et al.