a look at high-order finite- volume schemes for simulating atmospheric flows paul ullrich university...
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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)
1
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
2
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).
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