Local Time-Stepping for the Shallow Water Equations using CFL Optimized Forward-Backward Runge-Kutta Schemes

Abstract

The Courant-Friedrichs-Lewy (CFL) condition is a well known, necessary condition for the stability of explicit time-stepping schemes that effectively places a limit on the size of the largest admittable time-step for a given problem. We formulate and present a new local time-stepping (LTS) scheme optimized, in the CFL sense, for the shallow water equations (SWEs). This new scheme, called FB-LTS, is based on the CFL optimized forward-backward Runge-Kutta schemes from Lilly et al. (2023). We show that FB-LTS maintains exact conservation of mass and absolute vorticity when applied to the TRiSK spatial discretization (Ringler et al., 2010), and provide numerical experiments showing that it retains the temporal order of the scheme on which it is based (second order). In terms of computational performance, we show that when applied to a real-world test case on a highly-variable resolution mesh, the MPAS-Ocean implementation of FB-LTS is up to 10 times faster than the classical four-stage, fourth-order Runge-Kutta method (RK4), and 2.3 times faster than an existing strong stability preserving Runge-Kutta based LTS scheme (LTS3). Despite this significant increase in efficiency, the solutions produced by FB-LTS are qualitatively equivalent to those produced by both RK4 and LTS3.

Figures (10)

• Figure 1: Example TRiSK grid from a Voronoi tessellation, where the primal cells are hexagons and the dual cells are triangles centered at primal cell vertices. This is the type of spatial discretzation used by MPAS-Ocean. The vector $\mathbf{n}_e$ is normal to cell edge $e$ in a fixed, arbitrary direction. Later, in Section \ref{['subsec:conservation']}, we define a quantity $n_{e,i}$ as either 1 or -1 so that $n_{e,i} \mathbf{n}_e$ is the outward unit normal vector to cell $i$ at edge $e$. Then, $\mathbf{t}_e = \mathbf{k} \times \mathbf{n}_e$.
• Figure 2: An example mesh with cells and edges labeled for LTS. Blue cells and edges belong to $\mathcal{F}$, light blue cells and edges belong to $\mathcal{F}^\ell \subseteq \mathcal{F}$, pink cells and edges belong to $\mathcal{C}^\text{IF-1}$, yellow cells and edges belong to $\mathcal{C}^\text{IF-2}$, and red cells and edges belong to $\mathcal{C}^\text{int}$. Note that in practice, one often needs more layers of light blue, pink, and yellow cells; see Figure \ref{['fig:fblts_lts3_regions']} for a practical example.
• Figure 3: Visualization and notation for the time-levels used by FB-LTS. In the coarse region (illustrated in the right half of the diagram), $t^{n+1/m} = t^n + \frac{\Delta t}{m}$ for any positive integer $m$; first stage data is calculated at time $t^{n+1/3}$ and second stage data is calculated at time $t^{n+1/2}$. In the fine region (illustrated in the left half of the diagram), $t^{n,k} = t^n + k\frac{\Delta t}{M}$ and $t^{n,k+1/m} = t^n + \left(k+1/m \right)\frac{\Delta t}{M}$; first stage data is calculated at times $t^{n,k+1/3}$ and second stage data is calculated at times $t^{n,k+1/2}$ for $k = 0,\cdots, M-1$.
• Figure 4: Temporal convergence for FB-LTS on a test case consisting of an external gravity wave on a non-rotating aquaplanet. The time-steps on the horizontal axis are the time-steps used in the coarse region, in each case we have $M = 4$. Errors are computed against a reference solution generated by RK4 using a time-step of 10 s.
• Figure 5: TRiSK grid showing dual cells at the boundary between fine and interface one regions. Blue denotes the fine region and pink denotes the interface one region as in Figure \ref{['fig:lts_regions']}.

Theorems & Definitions (6)

• Theorem 1
• proof
• Theorem 2
• proof
• remark 1
• remark 2