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

Unstaggered vs. Staggered Grids

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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 1 Cell 2 Cell 3 Cell 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 ConstantMethod (PCoM)

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Cell 1 Cell 2 Cell 3 Cell 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 CubicMethod (PCM)

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Cell 1 Cell 2 Cell 3 Cell 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

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Cell 1 Cell 2

UL

UR

A crude choice of Riemann solver can result in excess diffusion, which can severely contaminate the solution.

The Riemann Solver

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Rusanov Riemann solver AUSM+-up Riemann solver

Results: Shallow Water Model

Williamson et al. (1992) Test Case 2 - Steady State Geostrophic Flow (=45)

Fluid Depth (h)Fluid Depth (h)

Results: Shallow Water Model

Results: Shallow Water ModelWilliamson et al. (1992) Test Case 5 - Flow over Topography

Total Fluid Depth (H)Total Fluid Depth (H)

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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 <# of vertical levels>.

• 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

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1,2,3. Explicit steps

4. Implicit step

1,3,5. Explicit steps

2,4. Implicit steps

Temperature at 500m

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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?paullric@umich.edu

http://www.umich.edu/~paullric

The Riemann solver introduces a natural source of damping, which can act to suppress oscillations in the divergence.

The Riemann Solver

2Advective Term

(proportional to dm/dx)

Diffusive Term(proportional to c dh4/dx4)

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

ShallowWater

Hydro-static