  This paper proposes and analyzes a novel efficient high-order finite volume
method for the ideal magnetohydrodynamics (MHD). As a distinctive feature, the
method simultaneously preserves a discretely divergence-free (DDF) constraint
on the magnetic field and the positivity-preserving (PP) property, which
ensures the positivity of density, pressure, and internal energy. To enforce
the DDF condition, we design a new discrete projection approach that projects
the reconstructed point values at the cell interface into a DDF space, without
using any approximation polynomials. This projection method is highly
efficient, easy to implement, and particularly suitable for standard high-order
finite volume WENO methods, which typically return only the point values in the
reconstruction. Moreover, we also develop a new finite volume framework for
constructing provably PP schemes for the ideal MHD system. The framework
comprises the discrete projection technique, a suitable approximation to the
Godunov--Powell source terms, and a simple PP limiter. We provide rigorous
analysis of the PP property of the proposed finite volume method, demonstrating
that the DDF condition and the proper approximation to the source terms
eliminate the impact of magnetic divergence terms on the PP property. The
analysis is challenging due to the internal energy function's nonlinearity and
the intricate relationship between the DDF and PP properties. To address these
challenges, the recently developed geometric quasilinearization approach is
adopted, which transforms a nonlinear constraint into a family of linear
constraints. Finally, we validate the effectiveness of the proposed method
through several benchmark and demanding numerical examples. The results
demonstrate that the proposed method is robust, accurate, and highly effective,
confirming the significance of the proposed DDF projection and PP techniques.
