A Look at High-Order FiniteVolume 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
• 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.
Why Finite Volumes?
• 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.
Unstaggered vs. Staggered Grids
• Many atmospheric models make use of staggered grids (ie.
Arakawa B,C,D-grids), where velocity components and massvariables 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
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Unstaggered vs. Staggered Grids
Finite Volume Formulation
• The high-order upwind finite volume model consists of several
components, a few of which will be covered here:
1
2
3
The sub-grid-scale reconstruction
The Riemann solver
The implicit-explicit dimension-split integrator
Sub-Grid Scale Reconstruction
1
Our sub-grid scale reconstruction can use only information on the cellaveraged values within each element.
Cell 1
Cell 2
Cell 3
Cell 4
Sub-Grid Scale Reconstruction
1
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.
Cell 1
Cell 2
Cell 3
Piecewise Constant
Method (PCoM)
Cell 4
Sub-Grid Scale Reconstruction
1
Increasing the accuracy of the method with respect to the
reconstruction simply requires using increasingly high order
polynomials for the sub-grid scale reconstruction.
Cell 1
Piecewise Cubic
Method (PCM)
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.
The Riemann Solver
2
Since the reconstruction is inherently discontinuous at cell interfaces,
we must solve a Riemann problem to obtain the flux of all conserved
variables.
UL
Cell 1
UR
Cell 2
The Riemann Solver
2
A crude choice of Riemann solver can result in excess diffusion, which
can severely contaminate the solution.
Rusanov Riemann solver
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AUSM+-up Riemann solver
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Results: Shallow Water Model
Williamson et al. (1992) Test Case 2 - Steady State Geostrophic Flow (=45)
Fluid Depth (h)
Results: Shallow Water Model
Results: Shallow Water Model
Williamson et al. (1992) Test Case 5 - Flow over Topography
Total Fluid Depth (H)
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Vertical Discretization
3
Vertically propagating sound waves are a major issue for
nonhydrostatic models. This suggests special treatment is required of
the vertical coordinate.
•
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 RungeKutta-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.
Vertical Discretization
3
Care must be taken to choose a high-order-accurate timestepping
scheme. Poor choices can lead to severely degraded model results.
1,2,3. Explicit steps
1,3,5. Explicit steps
4. Implicit step
2,4. Implicit steps
Results: 3D Nonhydrostatic Model
Jablonowski (2011) Baroclinic Instability in a Channel
Temperature at 500m
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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
2
The Riemann solver introduces a natural source of damping, which can
act to suppress oscillations in the divergence.
Example: Third-order reconstruction (parabolic sub-grid-scale) applied
to the linear shallow-water equations plus Riemann solver.
Advective Term
(proportional to dm/dx)
Diffusive Term
(proportional to c dh4/dx4)
Next Generation Climate Models
Advection
Finite Volume
High-order upwind
High-order symmetric
Compact Stencil
Discontinuous Galerkin
Spectral element / CG
Spectral volume
Semi-Lagrangian
Shallow
Water
Hydrostatic
Nonhydrostatic