A positivity-preserving adaptive-order finite-difference scheme for GRMHD

Abstract

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.