  The density and pressure are positive physical quantities in
magnetohydrodynamics (MHD). Design of provably positivity-preserving (PP)
numerical schemes for ideal compressible MHD is highly desirable, but remains a
challenge especially in the multidimensional cases. In this paper, we first
develop uniformly high-order discontinuous Galerkin (DG) schemes which provably
preserve the positivity of density and pressure for multidimensional ideal MHD.
The schemes are constructed by using the locally divergence-free DG schemes for
the symmetrizable ideal MHD equations as the base schemes, a PP limiter to
enforce the positivity of the DG solutions, and the strong stability preserving
methods for time discretization. The significant innovation is that we discover
and rigorously prove the PP property of the proposed DG schemes by using a
novel equivalent form of the admissible state set and very technical estimates.
Several two-dimensional numerical examples further confirm the PP property, and
demonstrate the accuracy, effectiveness and robustness of the proposed PP
methods.
