A Look at High-Order Finite- Volume Schemes for Simulating Atmospheric Flows Paul Ullrich University of Michigan

Next Generation Climate Models High-order accurate Move away from latitude-longitude grids Utilize modern hardware (GPUs, Petascale computing) Adaptive mesh refinement?

The cubed sphere grid is obtained by placing a cube inside the sphere and “inflating” it to occupy the total volume of the sphere. Pros: –Removes polar singularities –Grid faces are individually regular Cons –Some difficulty handling edges –Multiple coordinate systems Many atmospheric models now utilize this grid. The Cubed Sphere Grid

Finite volume methods have several advantages over finite difference and spectral methods: –They can be used to conserve invariant quantities, such as mass, energy, potential vorticity or potential enstrophy. –Finite volume methods can be easily made to satisfy monotonicity and positivity constraints (i.e. to avoid negative tracer densities). –Lots of research has been done on finite volume methods in aerospace and other CFD fields. Why Finite Volumes?

Many atmospheric models make use of staggered grids (ie. Arakawa B,C,D-grids), where velocity components and mass- variables are located at different grid points. Staggered grids have certain advantages, such as better treatment of high-wavenumber wave modes. However, staggered grids have stricter timestep constraints. Unstaggered grids allow us to easily perform horizontal-vertical dimension splitting. Staggered grids also suffer from unphysical wave reflection at abrupt grid resolution discontinuities (on adaptive grids)… Unstaggered vs. Staggered Grids

The high-order upwind finite volume model consists of several components, a few of which will be covered here: The sub-grid-scale reconstruction The Riemann solver The implicit-explicit dimension-split integrator Finite Volume Formulation 1 2 3

1 Our sub-grid scale reconstruction can use only information on the cell- averaged values within each element. Cell 1Cell 2Cell 3Cell 4 Sub-Grid Scale Reconstruction

The least accurate and least computation-intensive method for building a sub-grid scale reconstruction assumes that all points within a source grid element share the same value. Sub-Grid Scale Reconstruction Piecewise Constant Method (PCoM) 1 Cell 1Cell 2Cell 3Cell 4

Increasing the accuracy of the method with respect to the reconstruction simply requires using increasingly high order polynomials for the sub-grid scale reconstruction. Sub-Grid Scale Reconstruction Piecewise Cubic Method (PCM) 1 Cell 1Cell 2Cell 3Cell 4 A cubic reconstruction will lead to a 4th order accurate scheme, if paired with a sufficiently accurate timestep scheme.

Since the reconstruction is inherently discontinuous at cell interfaces, we must solve a Riemann problem to obtain the flux of all conserved variables. The Riemann Solver 2 Cell 1Cell 2 ULUL URUR

A crude choice of Riemann solver can result in excess diffusion, which can severely contaminate the solution. The Riemann Solver 2 Rusanov Riemann solverAUSM + -up Riemann solver

Results: Shallow Water Model Williamson et al. (1992) Test Case 2 - Steady State Geostrophic Flow (  =45  ) Fluid Depth (h)

Results: Shallow Water Model

Williamson et al. (1992) Test Case 5 - Flow over Topography Total Fluid Depth (H)

Vertically propagating sound waves are a major issue for nonhydrostatic models. This suggests special treatment is required of the vertical coordinate. Vertical Discretization 3 Idea: Since we are using an unstaggered grid, its easy to split the horizontal and vertical integration and treat the vertical integration implicitly, even in the presence of topography. Since vertical columns are disjoint, each column only requires a single implicit solve; total matrix size = 5 x. In order to achieve high-order accuracy we use Implicit-Explicit Runge- Kutta-Rosenbrock (IMEX-RKR) schemes. The resulting method is valid on all scales, uses the horizontal timestep constraint, is high-order accurate and is only modestly slower than a hydrostatic model.

Care must be taken to choose a high-order-accurate timestepping scheme. Poor choices can lead to severely degraded model results. Vertical Discretization 3 1,2,3. Explicit steps 4. Implicit step 1,3,5. Explicit steps 2,4. Implicit steps

Temperature at 500m Results: 3D Nonhydrostatic Model Jablonowski (2011) Baroclinic Instability in a Channel

Summary Next generation atmospheric models will likely rely on high-order numerical methods to achieve accuracy at a reduced computational cost. We have successfully demonstrated a high-order finite volume method for the shallow-water equations on the sphere and for nonhydrostatic 2D and 3D modeling. Implicit-explicit Runge-Kutta-Rosenbrock (IMEX-RKR) methods are very good candidates for time integrators, and can likely be adapted to any unstaggered grid model (high-order FV, DG, SV).

Questions?

The Riemann solver introduces a natural source of damping, which can act to suppress oscillations in the divergence. The Riemann Solver 2 Advective Term (proportional to dm/dx) Diffusive Term (proportional to c dh 4 /dx 4 ) Example: Third-order reconstruction (parabolic sub-grid-scale) applied to the linear shallow-water equations plus Riemann solver.

Next Generation Climate Models Finite Volume High-order upwind High-order symmetric Compact Stencil Discontinuous Galerkin Spectral element / CG Spectral volume Semi-Lagrangian Advection Nonhydro- static Shallow Water Hydro- static