  We propose a high-order finite difference weighted ENO (WENO) method for the
ideal magnetohydrodynamics (MHD) equations. The proposed method is
single-stage, single-step, maintains a discrete divergence-free condition on
the magnetic field, and has the capacity to preserve the positivity of the
density and pressure. To accomplish this, we use a Taylor discretization of the
Picard integral formulation (PIF) of the finite difference WENO method proposed
in [SINUM, 53 (2015), pp. 1833--1856], where the focus is on a high-order
discretization of the fluxes (as opposed to the conserved variables). We use
the version where fluxes are expanded to third-order accuracy in time, and for
the fluid variables space is discretized using the classical fifth-order finite
difference WENO discretization. We use constrained transport in order to obtain
divergence-free magnetic fields, which means that we simultaneously evolve the
magnetohydrodynamic and magnetic potential equations, and set the magnetic
field to be the (discrete) curl of the magnetic potential after each time step.
In this work, we compute these curls to fourth-order accuracy. In order to
retain a single-stage, single-step method, we develop a novel Lax-Wendroff
discretization for the evolution of the magnetic potential, where we start with
technology used for Hamilton-Jacobi equations in order to construct a
non-oscillatory magnetic field. Positivity preservation is realized by
introducing a parameterized flux limiter that considers a linear combination of
high and low-order numerical fluxes. This positivity limiter lacks energy
conservation. However, this limiter can be dropped for problems where the
pressure does not become negative. We present two and three dimensional
numerical results for several standard test problems. These results assert the
robustness and verify the high-order of accuracy of the proposed scheme.
