Abstract/Summary

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 iteration is needed each time the linear equation solver is called for solving the advection equation. The demonstration in this paper uses the second-order Runge-Kutta implicit/explicit time integration in combination with a second/third-order finite-volume spatial discretisation and is tested using deformation flow tracer advection on the sphere and a fully compressible model for atmospheric flows. Tracers are advected over the poles of highly anisotropic latitude-longitude grids with very large Courant numbers and on quasi-uniform hexagonal and cubed-sphere meshes with the same algorithm. Buoyant flow simulations with strong local updrafts also benefit from adaptively implicit advection. Stably stratified compressible flow simulations require a stable combination of implicit treatment of gravity and acoustic waves as well as advection in order to achieve long time-steps.