This paper describes a second‐order upwind scheme for multidimensional magnetohydrodynamics, which uses a linear approximation for all Riemann problems except those involving strong rarefactions. This enables it to cope with initial data for which previously published schemes might fail. The condition ▽⊙B = 0 is not enforced in multidimensions, but the numerical problems associated with this are dealt with by adding source terms to the equations, as suggested by Powell. We also show that there are advantages to adding second‐order artificial dissipation at shocks.

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