  Implicit time-stepping for advection is applied locally in space and time
where Courant numbers are large, but standard explicit time-stepping is used
for the remaining solution which is typically the majority. This adaptively
implicit advection scheme facilitates efficient and robust integrations with
long time-steps while having negligible impact on the overall accuracy, and
achieving monotonicity and local conservation on general meshes. A novel and
important aspect for the efficiency of the approach is that only one linear
solver iteration is needed for each advection solve.
  The implementation in this paper uses a second-order Runge-Kutta
implicit/explicit time-stepping in combination with a second/third-order finite
volume spatial discretisation. We demonstrate the adaptively implicit advection
in the context of deformational flow advection on the sphere and a fully
compressible model for atmospheric flows. Tracers are advected over the poles
of latitude-longitude grids with very large Courant numbers and through
hexagonal and cubed-sphere meshes with the same algorithm. Buoyant flow
simulations with strong local updrafts also benefit from adaptively implicit
advection. Stably stratified flow simulations require a stable combination of
implicit treatment of gravity and acoustic waves as well as advection in order
to achieve long stable time-steps.
