  In this paper, we extend the unsplit staggered mesh scheme (USM) for 2D
magnetohydrodynamics (MHD) (Lee and Deane, 2009) to a full 3D MHD scheme. The
scheme is a finite-volume Godunov method consisting of a constrained transport
(CT) method and an efficient and accurate single-step, directionally unsplit
multidimensional data reconstruction-evolution algorithm, which extends the
original 2D corner transport upwind (CTU) method (Colella, 1990). We present
two types of data reconstruction-evolution algorithms for 3D: (1) a reduced CTU
scheme and (2) a full CTU scheme. The reduced 3D CTU scheme is a variant of a
simple 3D extension of the 2D CTU method by Colella (1990) and is considered as
a direct extension from the 2D USM scheme. The full 3D CTU scheme is our
primary 3D solver which includes all multidimensional cross-derivative terms
for stability. The latter method is logically analogous to the 3D unsplit CTU
method by Saltzman. The major novelties in our algorithms are twofold. First,
we extend the reduced CTU scheme to the full CTU scheme which is able to run
with CFL numbers close to unity. Both methods utilize the transverse update
technique developed in the 2D USM algorithm to account for transverse fluxes
without solving intermediate Riemann problems, which in turn gives
cost-effective 3D methods by reducing the total number of Riemann solves. The
proposed algorithms are simple and efficient especially when including
multidimensional MHD terms that maintain in-plane magnetic field dynamics.
Second, we introduce a new CT scheme that makes use of proper upwind
information in taking averages of electric fields.
