Implicit and semi-implicit schemes: Algorithms

This study formulates general guidelines to extend an explicit code with a great variety of implicit and semi-implicit time integration schemes. The d iscussion is based on their specific implementation in the Versatile Advect ion Code, which is a general purpose software package for solving systems o f non-linear hyperbolic (and/or parabolic) partial differential equations, using standard high resolution shock capturing schemes. For all combination s of explicit high resolution schemes with implicit and semi-implicit treat ments, it is shown how second-order spatial and temporal accuracy for the s mooth part of the solutions can be maintained. Strategies to obtain steady state and time accurate solutions implicitly are discussed. The implicit an d semi-implicit schemes require the solution of large linear systems contai ning the Jacobian matrix. The Jacobian matrix itself is calculated numerica lly to ensure the generality of this implementation. Three options are disc ussed in terms of applicability, storage requirements and computational eff iciency. One option is the easily implemented matrix-free approach, but the Jacobian matrix can also be calculated by using a general grid masking alg orithm, or by an efficient implementation for a specific Lax-Friedrich-type total variation diminishing (TVD) spatial discretization. The choice of th e linear solver depends on the dimensionality of the problem. In one dimens ion, a direct block tridiagonal solver can be applied, while in more than o ne spatial dimension, a conjugate gradient (CG)-type iterative solver is us ed. For advection-dominated problems, preconditioning is needed to accelera te the convergence of the iterative schemes. The modified block incomplete LU-preconditioner is implemented, which performs very well. Examples from t wo-dimensional hydrodynamic and magnetohydrodynamic computations are given. They model transonic stellar outflow and recover the complex magnetohydrod ynamic bow shock flow in the switch-on regime found in De Sterck et al. Cop yright (C) 1999 John Wiley & Sons, Ltd.