Adaptively implicit MPDATA advection for arbitrary Courant numbers and meshes

This can lead to instability in the presence of high flow speeds (relative to mesh spacing) such as occurs in convective updraughts, regions of mesh convergence, or where the winds are unusually high. An adaptively implicit advection scheme is proposed that treats advection implicitly only where the Courant number is high.

Flux correction to ensure monotonicity is adapted to work with implicit time stepping. A version of the multidimensional positive-definite advection transport algorithm MPDATA is derived with an anti-diffusive flux compensating truncation errors of off-centred implicit time stepping. The anti-diffusive flux is gradually reduced as Courant numbers increase above 2 in order to maintain stability at the expense of second-order accuracy at high Courant numbers.