In this paper an approach to multidimensional magnetohydrodynamics (MI-ID)
which correctly handles shocks but does not use an approximate Riemann solv
er is proposed. This approach is simple and is based on control volume aver
aging with a staggered grid. The method builds on the older and often overl
ooked technique of on each step taking a fully 3-D Lagrangian step and then
conservatively remapping onto the original grid. At the remap step gradien
t limiters are applied so that the scheme is monotonicity-preserving. For E
uler's equations this technique, combined with an appropriately staggered g
rid and Wilkins artificial viscosity, can give results comparable to those
from approximate Riemann solvers. We show how this can be extended to inclu
de a magnetic field, maintaining the divergence-free condition and pressure
positivity and then present numerical test results. Where possible a compa
rison with other shock capturing techniques is presented and the advantages
and disadvantages of the proposed scheme are clearly explained. (C) 2001 A
cademic Press.