AdHImEx: Adaptively High-order Implicit-Explicit Time Stepping for Atmospheric Transport with Long Time Steps

Weather and climate models require atmospheric transport to be numerically stable, accurate, efficient, and mass-conserving. Adaptively Implicit-Explicit (AdImEx) time stepping can provide mass conservation and stability while taking long time steps for efficiency. However, challenges persist: (1) strongly reduced accuracy for large Courant numbers, and (2) a lack of preservation of uniform fields with substage time step sizes varying in space. We introduce AdHImEx, a novel AdImEx approach that resolves these issues, delivering an unconditionally stable, conservative, efficient advection scheme that preserves unity and increases temporal accuracy for large Courant numbers from first- to second-order. Remarkably, it requires only a single implicit solve per time step, performed with an iterative matrix solver, while retaining the full efficiency and third-order accuracy of the explicit time stepping in regions with small Courant numbers. Two-dimensional results will be shown.