Adaptively implicit MPDATA advection for arbitrary Courant numbers and Meshes

Advection schemes with time step restrictions are widely used in weather and climate models. This can lead to instability in the presence of high flow speeds (relative to mesh spacing) such as occurs in convective updrafts, regions of mesh convergence or where the winds are unusually high. An adaptively implicit advection scheme is proposed which 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 MPDATA (Multidimensional Positive Definite Advection Transport Algorithm) 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. Results of two-dimensional advection by deformational flow are presented on various meshes of the sphere. Stability and second-order accuracy are maintained when the Courant number is over 100 in a small region, when strong wind crosses the poles of a rotated latitude-longitude mesh. Good solutions are also obtained on a skipped latitude-longitude mesh, a cubed sphere and hexagonal meshes. Accuracy reverts to first-order when Courant numbers are large over a large fraction of the domain