This paper introduces a previously unaddressed problem with performing
multidimensional numerical MHD characterized by ''explosive'' growth
(within a single time step) of weak magnetic fields to possibly dynami
c strengths in the vicinity of strong velocity shear. This problem has
been observed to occur in numerical algorithms which rely on the solu
tion of the characteristic equations at the mesh interfaces. The solut
ion to this problem is the subject of this paper and has led to an alg
orithm fundamentally different from those discussed previously. Specif
ically, at every point on the mesh, the magnetic induction terms are d
etermined from the characteristic velocities which are evaluated by so
lving implicitly (in space) the characteristic equations for incompres
sible MHD (although the algorithm is perfectly suitable for compressib
le MHD). The implicit treatment of the characteristic equations sets t
his algorithm apart from previous efforts and is shown to be a necessa
ry condition for multidimensional accuracy. It amounts to replacing th
e usual directional-split treatment of the induction and transport ope
rators with a planar-split scheme.