  Balsara (2001, J. Comput. Phys., 174, 614) showed the importance of
divergence-free reconstruction in adaptive mesh refinement problems for
magnetohydrodynamics (MHD) and the importance of the same for designing robust
second order schemes for MHD was shown in Balsara (2004, ApJS, 151, 149).
Second order accurate divergence-free schemes for MHD have shown themselves to
be very useful in several areas of science and engineering. However, certain
computational MHD problems would be much benefited if the schemes had third and
higher orders of accuracy. In this paper we show that the reconstruction of
divergence-free vector fields can be carried out with better than second order
accuracy. As a result, we design divergence-free weighted essentially
non-oscillatory (WENO) schemes for MHD that have order of accuracy better than
second. A multistage Runge-Kutta time integration is used to ensure that the
temporal accuracy matches the spatial accuracy. Accuracy analysis is carried
out and it is shown that the schemes meet their design accuracy for smooth
problems. Stringent tests are also presented showing that the schemes perform
well on those tests.
