Finite volume transport schemes for Voronoi meshes

Finite volume transport schemes
for Voronoi meshes
Bill Skamarock NCAR/NESL/MMM
Scalar transport cost can be a large part of the dry-dynamics cost:
NWP (high-res: < 10 km Dx) – O(10) scalars.
Climate – O(30) scalars (aerosols and trace gases).
Chemistry/air quality – O(100) scalars.
Atmospheric modeling experience: Overall solution accuracy is
strongly dependent on accuracy of scalar transport.

Conservation of mass
(conservative, flux form)
Conservation of scalar mass
(conservative, flux form)
Finite-volume formulation:
Integrating in space and time
over a fixed volume,
use the divergence theorem
where f is a dimensionless mixing ratio
Consider two formulations – (1)
Scheme: approximate space-time integral of flux through cell surface

Consider two formulations – (2)
Finite-volume formulation:
Integrating in space over a
fixed volume, use the
divergence theorem
Scheme: instantaneous integral of flux through cell surface, ODE (time)

MPAS (Voronoi mesh): Formulation (2)
Instantaneous
flux divergence in
RK-based scheme

MPAS (Voronoi mesh): Formulation (2)
Computing the flux - consider 1D transport (e.g. from WRF)
2nd-order flux:
3rd and 4th-order fluxes: (Hundsdorfer et al, JCP 1995; van Leer, 1985)
where

MPAS (Voronoi mesh): Formulation (2)
Recognizing
We recast the 3rd and 4th order flux
for the hexagonal grid as
where x is the direction normal to the cell edge and i and i+1 are cell centers.
We use the least-squares-fit polynomial to compute the second derivatives.

MPAS (Voronoi mesh): Formulation (2)
Edge e 1 has weights for
computing second derivatives at
cell centers C 0 and C 1.
The weights for C 0 apply to cell
centers C 0 through C 6, and the
weights for C 1 apply to cell
centers C 0-C 2 and C 6-C 9.

MPAS (Voronoi mesh): Formulation (2)

MPAS (Voronoi mesh): Formulation (2)
Day 9 solution,
Jablonowski and Williamson (2006)
baroclinic wave test case
10242 cell (240 km) mesh solution errors

MPAS (Voronoi mesh): Formulation (1)
Finite-volume formulation:
Integrating in space and time
over a fixed volume,
use the divergence theorem
Scheme: approximate space-time integral of flux through cell surface

MPAS (Voronoi mesh): Formulation (1)

MPAS (Voronoi mesh): Formulation (1)
M07 and LR05 used first-order reconstructions
Lashley and Thuburn (2002) and
Skamarock and Menchaca (2010)
use second and fourth-order reconstructions, e.g.
The least-squares polynomial fit is constrained to pass through the cell-center value
by fitting a polynomial of the form
The constant (c 0 = � 0) is adjusted such that the cell-integrated polynomial is equal
to the cell-average value times the cell area.

MPAS (Voronoi mesh): Formulations (1) and (2)
Blossey and Durran
deformational flow
test case

MPAS (Voronoi mesh): Formulation (1*)
(1) Fit quadratic polynomial to cell
center and cell vertex values
(2) Cell-vertex values are a weighted
sum of nearest and next-nearest cells

MPAS (Voronoi mesh): Formulation (1*)
(3) Polynomial constraints
(2) Cell-vertex values
Perfect hexagons: = 3
2
This leads to a 4 th order accurate
gradient operator on perfect hexagons
1
2

MPAS (Voronoi mesh): Formulations (1*) and (2)

MPAS (Voronoi mesh): Formulations (1) and (1*)
Solid body rotation, slotted cylinder

MPAS (Voronoi mesh): Formulations (2) and (1*)
Slotted cylinder deformational flow test case
Lauritzen et al (2012)
Miura and Skamarock (2012)
formulation (1*)
FCT limiter
T/2 T
Skamarock and Gassmann (2011)
formulation (2)
FCT limiter
Miura and Skamarock (2012); formulation (1*)
Skamarock and Gassmann (2011); formulation (2)

MPAS (Voronoi mesh): Formulations (1*)
Blossey and Durran deformational flow test case
Blossey and Durran
deformational flow
test case
Dx 60 km Dx 60 km, FCT lim Dx 120 km, FCT lim Dx 240 km, FCT lim

MPAS (Voronoi mesh): Formulations (1*)
Blossey and Durran deformational flow test case
MS (2012) FIT limiter
Dx = 120 km
MS (2012) FIT
(lower bnd only)
Dx = 120 km
MS (2012) FIT
(lower bnd only)
Dx = 60 km

Finite volume transport schemes
for Voronoi meshes
Summary
We examined 3 newer schemes using 2+ order reconstructions or
higher-order formulations.
The new schemes show significantly reduced phase and amplitude
error relative to 1 st order reconstructions.
Runge-Kutta-based scheme appears to be the most robust.
While Miura and Skamarock (2012) scheme appears most accurate,
some problematic behavior (non-convergence) appears when used with
FCT-type limiters.
Nonlinear limiters/renormalization – lessons learned?