Adaptively Implicit Advection for Atmospheric Flows

TL;DR

This work tackles time-step limitations in atmospheric transport by introducing adaptively implicit advection within a two-stage IMEX Runge–Kutta framework. By applying implicit stepping only where the local Courant number is large and retaining explicit updates elsewhere, the method achieves monotonicity, local conservation, and unconditional stability for the advection step, with only one linear solver iteration per application. The approach integrates a quasi-cubic upwind spatial discretisation in OpenFOAM and couples advection with implicit treatment of gravity and acoustic waves through a Helmholtz pressure solve, demonstrated across deformational flow on spheres, buoyancy-driven flows, and stably stratified cases on diverse meshes. The results show substantial cost savings at large time-steps without sacrificing accuracy in regions with small Courant numbers, making the method promising for scalable, realistic weather and climate simulations and enabling longer-time integrations on complex meshes.

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.

Figures (18)

• Figure 1: Cell $C$, with neighbour $N$, straddling face $f$ in an arbitrary mesh. Cells have centres $\mathbf{x}_{C}$ and $\mathbf{x}_{N}$ and face $f$ has centre $\mathbf{x}_{f}$. Cell centre quantity $\psi$ has value $\psi_{f}$ at the face. Cells have volume $\mathcal{V}$. Faces have area vectors $\mathbf{S}_{f}$ normal to the face with magnitude equal to the face area.
• Figure 2: Numerical analysis of implicit/explicit cubic upwind solution of (\ref{['eq:1dgw']}). In each panel, the maximum of the magnitude of amplification factor $|\mathcal{A}|$ over all wavenumbers, ${k\Delta x\in\left[0,2\pi\right]}$ is plotted ($\max_{k\Delta x}|\mathcal{A}|$). Top left: $\max_{k\Delta x}|\mathcal{A}|$ as a function of $c$ for $N\Delta t=0$ for schemes with various combinations of $\alpha$, $\beta$ and $\gamma$. The other panels show $\max_{k\Delta x}|\mathcal{A}|$ contoured as a function of $c$ and $N\Delta t$ with $|\mathcal{A}|=1$ contoured black, again for different combinations of $\alpha$, $\beta$ and $\gamma$. The formulation for $\alpha$ in the bottom right is unconditionally stable so there is no contour for $|\mathcal{A}|=1$.
• Figure 3: Deformational flow on the sphere advecting Gaussian hill shaped distributions of $\psi$ at the half-time 2.5 using $\gamma=1$. Dashed contours at $\psi=-10^{-2},\ -2\times10^{-2},\ -3\times10^{-2}$.
• Figure 4: Error norms for deformational flow advection of Gaussian hill shaped distributions. The left panel shows errors as a function of spatial resolution while keeping the time-step proportional. The right panel shows errors as a function of time-step on the C60 cubed sphere. $\gamma=1$ unless otherwise stated.
• Figure 5: Final results of deformational flow advection of slotted cylinders on the sphere.