Adaptively Implicit Advection for Atmospheric Flows

TL;DR

This paper introduces an adaptively implicit advection scheme that combines explicit and implicit methods to enable efficient, stable, and accurate atmospheric flow simulations with large time-steps on complex meshes.

Contribution

The paper presents a novel adaptively implicit advection method that requires only one linear solver iteration per solve, improving efficiency and stability in atmospheric flow modeling.

Findings

Effective advection over poles and complex meshes with large Courant numbers.

Stable long-time integration of buoyant and stratified flows.

Reduced computational cost due to single-iteration linear solves.

Abstract

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.