  Numerical schemes provably preserving the positivity of density and pressure
are highly desirable for MHD, but the rigorous positivity-preserving (PP)
analysis remains challenging. The difficulties mainly arise from the intrinsic
complexity of the MHD equations as well as the indeterminate relation between
the PP property and the divergence-free condition on magnetic field. We present
the first rigorous PP analysis of conservative schemes with Lax-Friedrichs (LF)
flux for ideal MHD. The significant innovation is the discovery of theoretical
connection between PP property and a discrete divergence-free (DDF) condition.
This connection is established through the generalized LF splitting properties,
which are alternatives of the usually-expected LF splitting property that does
not hold for ideal MHD. The generalized LF splitting properties involve a
number of admissible states strongly coupled by DDF condition, making their
derivation very difficult. We derive these properties via a novel equivalent
form of the admissible state set and an important inequality skillfully
constructed by technical estimates. Rigorous PP analysis is presented for
finite volume and discontinuous Galerkin schemes with LF flux on uniform
Cartesian meshes. In 1D case, PP property is proved for the first-order scheme
with proper numerical viscosity, and also for arbitrarily high-order schemes
under conditions accessible by a PP limiter. In 2D case, we show that the DDF
condition is necessary and crucial for achieving PP property. It is observed
that even slightly violating the proposed DDF condition may cause failure to
preserve the positivity of pressure. We prove that the 2D LF type scheme with
proper numerical viscosity preserves the positivity and DDF condition.
Sufficient conditions are derived for 2D PP high-order schemes, and extension
to 3D is discussed. Numerical examples confirm the theoretical findings.
