  We present an adaptive-order positivity-preserving conservative
finite-difference scheme that allows a high-order solution away from shocks and
discontinuities while guaranteeing positivity and robustness at
discontinuities. This is achieved by monitoring the relative power in the
highest mode of the reconstructed polynomial and reducing the order when the
polynomial series no longer converges. Our approach is similar to the
multidimensional optimal order detection (MOOD) strategy, but differs in
several ways. The approach is a priori and so does not require retaking a time
step. It can also readily be combined with positivity-preserving flux limiters
that have gained significant traction in computational astrophysics and
numerical relativity. This combination ultimately guarantees a physical
solution both during reconstruction and time stepping. We demonstrate the
capabilities of the method using a standard suite of very challenging 1d, 2d,
and 3d general relativistic magnetohydrodynamics test problems.

High Energy Astrophysical Phenomena;General Relativity and Quantum Cosmology