  In this work we present a general strategy for constructing multidimensional
Riemann solvers with a single intermediate state, with particular attention
paid to detailing the two-dimensional Riemann solver. This is accomplished by
introducing a constant resolved state between the states being considered,
which introduces sufficient dissipation for systems of conservation laws.
Closed form expressions for the resolved fluxes are also provided to facilitate
numerical implementation. The Riemann solver is proved to be positively
conservative for the density variable; the positivity of the pressure variable
has been demonstrated for Euler flows when the divergence in the fluid
velocities is suitably restricted so as to prevent the formation of cavitation
in the flow.
  We also focus on the construction of multidimensionally upwinded electric
fields for divergence-free magnetohydrodynamical flows. A robust and efficient
second order accurate numerical scheme for two and three dimensional Euler and
magnetohydrodynamic flows is presented. The scheme is built on the current
multidimensional Riemann solver. The number of zones updated per second by this
scheme on a modern processor is shown to be cost competitive with schemes that
are based on a one-dimensional Riemann solver. However, the present scheme
permits larger timesteps.
