Compatible finite element methods for numerical
weather prediction
Colin Cotter (Mathematics, Imperial College London)
Wednesday 17th March 2015
Colin Cotter
Compatible FEM
The story so far
Aim of GungHo project (Met Office/NERC/STFC)
To research, design and develop a new dynamical core suitable for
operational, global and regional, weather and climate simulation on
massively parallel computers of the size envisaged over the coming
20 years.
Our goal
To find a set of numerical discretisations that enjoy the same
properties as the staggered finite difference method (C-grid), but
that can be used on pseudo-uniform grids.
Colin Cotter
Compatible FEM
• Pole singularities require special filtering
• Polar filters do not scale well on massively parallel computers
“overset grids.” There •have
alsoanisotropic
been recent
to use
grids based on octahedro
Highly
gridattempts
cells at high
latitudes
McGregor, 1996; Purser and Rancic, 1998). A “Fibonacci grid” has also been s
(Swinbank and Purser, 2006).
Compatible FEM
Consideration of alternative spatial discretizations:
Lat-Lon
Icosahedral-triangles
Icosahedral-hexagons
Cubed Sphere
Yin-Yan
FEEC or compatible/mimetic finite elements
Priority Requirements: •
Efficient on existing and proposed supercomputer architect
Fig. 11.8: Various ways of discretizing the sphere. Figure made by Bill Skamarock of NCA
well on massively
computers
Finite
element spaces built •asScales
a discrete
de Rhamparallel
complex.
• Well suited for cloud (nonhydrostatic) to global scales
1. No orthogonality constraints
on for
mesh.
• Capability
local grid refinement and regional domains
MPAS Grids based on icosahedra
offer
an
for
simulation of th
•
Conserves
atattractive
leastDOF
massframework
and
scalar
2. Flexibility to change pressure/velocity
ratios
toquantities
avoid
circulation of the atmosphere; their advantages include almost uniform and quasispurious mode branches.
Mesoscale & Microscale Meteorology Division / ESS
resolution over the sphere. Such grids are
termed “geodesic,” because they resemble the
3.
Flexibility
to
increase
consistency
order. (1968) and Sadourny (1968) simult
domes designed by Buckminster Fuller. Williamson
introduced a new approach to more homogeneously discretize the sphere. They construc
spherical triangles which are equilateral and nearly equal in area. Because the gr
are not regularly spaced and do not lie in orthogonal rows and columns, alternativ
using
Colin Cotter
Compatible FEM
Compatible finite element spaces
∇⊥ =(−∂y ,∂x )
H 1 ÐÐÐÐÐÐÐ→ H(div) ÐÐÐ→ L2
×
×
×
×π2
×π1
×π0
×
×
×
Ö
Ö
Ö
V0
∇⊥
ÐÐÐ→
∇⋅
V1
ÐÐÐ→ V2
∇⋅
▸
Arnold, Falk and Winther, Acta Numerica (2006).
▸
CJC and Shipton, JCP (2012). [steady geostrophic modes]
▸
Thuburn and CJC, JCP (online, in press). [dual grid]
Colin Cotter
Compatible FEM
∇⊥
V0 ÐÐÐ→ V1 ÐÐÐ→ V2
Q1→RT0→Q0
∇⋅
Q2→RT1→Q1DG
P2+B3→BDFM1→P1DG
Degree of freedom ratio condition, CJC and Shipton (2012)
There will be spurious branches in the numerical dispersion relation
if dim(V1 ) ≠ 2 dim(V2 ) in the periodic plane.
Colin Cotter
Compatible FEM
Dual operators and Helmholtz
∇⊥
∇⋅
˜ ⊥⋅
∇
←ÐÐÐ
V0 ÐÐÐ→ V1 ÐÐÐ→ V2
←ÐÐÐ
˜
∇
˜ ⊥ ⋅ u⟩ = −⟨∇⊥ γ, u⟩, γ ∈ V0 ,
⟨γ, ∇
˜ = −⟨∇ ⋅ w, φ⟩, ∀w ∈ V1 .
⟨w, ∇φ⟩
We have
˜ ⊥ ⋅ ∇φ⟩
˜ = −⟨∇⊥ ⋅ γ, ∇φ⟩
˜ = ⟨∇ ⋅ ∇⊥ γ , φ⟩ = 0,
⟨γ, ∇
´¹¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¶
=0
˜ ⊥ ⋅ ∇φ
˜ = 0 for all φ ∈ V2 .
so ∇
Colin Cotter
Compatible FEM
∀γ ∈ V0 .
Helmholtz decomposition
Helmholtz decomposition
˜ 2 ⊕ Hδ
V1 = ∇⊥ V1 ⊕ ∇V
˜ ⊥ ⋅ u = 0}
where Hδ = {u ∈ V1 ∶ ∇ ⋅ u = 0, ∇
δ
˜ + h , ψ ∈ V0 , φ ∈ V2 , hδ ∈ Hδ
i.e., u ∈ V1 Ô⇒ u = ∇⊥ ψ + ∇φ
Orthogonality
˜ = ⟨∇⊥ ψ, hδ ⟩ = ⟨∇φ,
˜ hδ ⟩ = 0, ∀ ψ ∈ V0 , φ ∈ V2 , hδ ∈ Hδ
⟨∇⊥ ψ, ∇φ⟩
Proposition
Arnold, Falk and Winther (2006) Under very weak approximation
assumptions, Hδ has the same dimension as H.
Colin Cotter
Compatible FEM
Linear properties
Linearised equations (barotropic case):
ut + f u⊥ +
°
Coriolis
∇Φ
°
= 0, ,
(u⊥ = n × u),
Pressure gradient
Φt + Φ0 ∇ ⋅ u = 0.
´¹¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹¶
Mass flux
Mixed finite element discretisation: seek u ∈ V1 , Φ ∈ V2 s.t.
Colin Cotter
Compatible FEM
⟨w, ut ⟩ + f ⟨w, u⊥ ⟩ − ⟨∇ ⋅ w, Φ⟩ = 0,
∀w ∈ V1 ,
⟨γ, Φt ⟩ + ⟨γ, Φ0 ∇ ⋅ u⟩ = 0,
∀γ ∈ V2 .
Linear properties
Mixed finite element discretisation: seek u ∈ V1 , Φ ∈ V2 s.t.
⟨w, ut ⟩ + f ⟨w, (u)⊥ ⟩ − ⟨∇ ⋅ w, Φ⟩ = 0,
∀w ∈ V1 ,
(1)
⟨γ, Φt ⟩ + ⟨γ, Φ0 ∇ ⋅ u⟩ = 0,
∀γ ∈ V2 .
(2)
For weather forecasting applications, would like to reproduce the
following C-grid properties:
▸
No spurious pressure modes (inf-sup condition) [classical].
▸
Geostrophic balance condition.
Colin Cotter
Compatible FEM
Proposition (CJC and Shipton (2012))
Let V0 , V1 , V2 be as above.
For all u ∈ ∇⊥ V0 ⊂ V1 , there exists Φ ∈ V0 such that (u, Φ) is a
steady state solution of equations (1) and (2).
Proof.
Take ψ ∈ V0 with ∇⊥ ψ = u, define Φ ∈ V2 by ⟨γ, f Ψ⟩ = ⟨γ, Φ⟩,
∀γ ∈ V2 .
⟨w, ut ⟩ = −⟨w, f (u)⊥ ⟩ + ⟨∇ ⋅ w, Φ⟩,
= ⟨w, f ∇ψ⟩ + ⟨∇ ⋅ w, Φ⟩,
= −⟨∇ ⋅ w, f ψ⟩ + ⟨∇ ⋅ w, Φ⟩ = 0, ∀w ∈ V1 .
Colin Cotter
Compatible FEM
Inertial oscillations
▸
For u ∈ Hδ , we have
ut + f u⊥ = 0,
i.e., u undergoes inertial oscillations.
▸
▸
Hδ has the same dimension as H, so there are the same
number of inertial modes as for the undiscretised equations.
For example, in the periodic plane, there are 2 independent
inertial modes, and hence no spurious ones.
Colin Cotter
Compatible FEM
Nonlinear shallow water equations
1
ut + qDu⊥ + ∇ (gD + ∣u∣2 ) = 0,
2
Dt + ∇ ⋅ (uD) = 0,
where q =
∇⊥ ⋅ u + f
.
D
Discretisation: u ∈ V1 , D ∈ V2 , and define F ∈ V1 , q ∈ V0 with
⟨w, F⟩ = ⟨w, uD⟩,
∀w ∈ V1 ,
⟨γ, qD⟩ = ⟨−∇ γ, u⟩ + ⟨γ, f ⟩,
1
⟨w, ut ⟩ + ⟨w, qF⊥ ⟩ − ⟨∇ ⋅ w, gD + ∣u∣2 ⟩ = 0, ∀w ∈ V1 ,
2
⟨φ, Dt + ∇ ⋅ F⟩ = 0, ∀φ ∈ V2 .
⊥
D equation is satisfied pointwise!
Colin Cotter
Compatible FEM
∀γ ∈ V0 ,
Energy conservation
E =∫
1
1
D∣u∣2 + gD 2 dx,
2
2
1
Ė = ⟨ ∣u∣2 + gD, Dt ⟩ + ⟨Du, ut ⟩,
2
´¹¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹¶
=⟨F,ut ⟩
1
= ⟨ ∣u∣2 + gD, −∇ ⋅ F⟩
2
1
+ ⟨F, −qF⊥ ⟩ + ⟨∇ ⋅ F, ∣u∣2 + gD⟩ = 0.
2
Colin Cotter
Compatible FEM
Z = ∫ Dq 2 dx,
Ż = −⟨Ḋ, q 2 ⟩ + 2⟨q, (qD)t ⟩,
= −⟨Ḋ, q 2 ⟩ − 2⟨∇⊥ q, ut ⟩,
= ⟨∇ ⋅ F, q 2 ⟩ + 2⟨∇⊥ q, qF⊥ ⟩
− 2⟨∇ ⋅ ∇⊥ q , qF⊥ ⟩,
´¹¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¶
=0
= −⟨F, ∇q 2 ⟩ + ⟨∇q 2 , F⟩ = 0.
Stabilisation schemes (e.g. SUPG) for q replace qF⊥ by q̃F⊥ ,
dissipating enstrophy whilst preserving energy.
Colin Cotter
Compatible FEM
Implied PV conservation
1
⟨w, ut ⟩ + ⟨w, qF⊥ ⟩ − ⟨∇ ⋅ w, gD + ∣u∣2 ⟩ = 0,
2
For γ ∈ V0 , take w = −∇⊥ γ,
⟨−∇⊥ γ, ut ⟩ − ⟨∇γ, qF⟩ = 0,
∀w ∈ V1 .
∀γ ∈ V0 .
Recall definition of q:
⟨γ, qD⟩ = ⟨−∇⊥ γ, u⟩ + ⟨γ, f ⟩,
∀γ ∈ V0 .
Hence we get the implied equation for diagnostic q,
⟨γ, (qD)t ⟩ − ⟨∇γ, qF⟩ = 0,
Colin Cotter
Compatible FEM
∀γ ∈ V0 .
Colin Cotter
Compatible FEM
Modified scheme:
⟨w, ut ⟩ + ⟨w, F⊥ (q − α ((qD)t + ∇ ⋅ (Fq)))⟩
1
−⟨∇ ⋅ w, gD + ∣u∣2 ⟩ = 0, ∀w ∈ V1 .
2
⟨γ, (qD)t ⟩ − ⟨∇γ, qF − αF ((qD)t + ∇ ⋅ (qF))⟩ = 0, ∀γ ∈ V0 .
Get SUPG discretisation for implied conservation law for q,
⟨γ + αF ⋅ ∇γ, (qD)t ⟩ − ⟨∇γ, qF⟩ + ⟨αF ⋅ ∇γ, ∇ ⋅ (qF)⟩ = 0,
▸
▸
▸
∀γ ∈ V0 .
Energy is still conserved but enstrophy is dissipated.
Approximation is still consistent at expected order.
Adding nonlinear limiters to this equation leads to dissipation
of energy too.
Colin Cotter
Compatible FEM
Colin Cotter
Compatible FEM
Normalised L2 error
10-1
10-2
10-3
10-4
10-5 5
10
Colin Cotter
Compatible FEM
RT1
BDM1
BDFM2
BDM2
∝ h1
∝ h2
106
h (m)
107
More nonlinear shallow water equations
▸
▸
▸
▸
▸
FEniCS tools for implementation on the sphere Rognes, Ham,
CJC and McRae GMD (2013).
Energy-enstrophy conserving formulation McRae and CJC
QJRMS (2014).
Connections with finite element exterior calculus CJC and
Thuburn JCP (2014).
Dual-grid, PV conservation Thuburn and CJC JCP (in press).
Another paper in preparation with Jemma Shipton on
incorporation of stable advection schemes.
Colin Cotter
Compatible FEM
Next steps
∇⊥
Given 2D finite element spaces (U0 → U1 → U2 ) and 1D finite
∇⋅
∂x
element spaces (V0 → V1 ), we can generate a product in three
dimensions:
∇
∇×
∇⋅
W0 Ð→ W1 Ð→ W2 Ð→ W3 ,
where
W0 ∶= U0 ⊗ V0 ,
W1 ∶= HCurl(U0 ⊗ V1 ) ⊕ HCurl(U1 ⊗ V0 ),
W2 ∶= HDiv(U1 ⊗ V1 ) ⊕ HDiv(U2 ⊗ V0 ),
W3 ∶= U2 ⊗ V1 ,
with W0 ⊂ H 1 , W1 ⊂ H(curl), W2 ⊂ H(div), W3 ⊂ L2
Colin Cotter
Compatible FEM
Density space
Colin Cotter
Compatible FEM
Vertical part of velocity space
Colin Cotter
Compatible FEM
Horizontal part of velocity space
Tensor product construction of FEEC spaces: Arnold, Boffi, and
Bonizzoni Numer. Math. (2015)
Symbolic algebra and Firedrake implementation: McRae, Bercea,
Mitchell, Ham and CJC (submitted, on arXiv)
Colin Cotter
Compatible FEM
Building finite element spaces in physical domain
X1
GK
(0, 1)
x
X
(0, 0)
(1, 0) X0
Given finite element spaces on reference element K̂ , with
∇⊥
V0 (K̂ ) ÐÐÐ→ V1 (K̂ ) ÐÐÐ→ V2 (K̂ )
we want transformations to spaces on physical elements K , with
∇⊥
∇⋅
V0 (K ) ÐÐÐ→ V1 (K ) ÐÐÐ→ V2 (K ),
∇⋅
plus inter-element continuity requirements.
Colin Cotter
Compatible FEM
Building finite element spaces in physical domain
X1
GK
(0, 1)
x
X
(0, 0)
(1, 0) X0
Pullbacks to reference element (2D):
▸
ψ ∈ V0 (K ): ψ ○ GK = ψ̂, for ψ̂ ∈ V0 (K̂ ).
▸
u ∈ V1 (K ): u ○ GK = J û/ det J, for û ∈ V1 (K̂ ).
▸
ρ ∈ V2 (K ): ρ ○ GK = ρ̂/ det J, for ρ̂ ∈ V2 (K̂ ).
Colin Cotter
Compatible FEM
Building finite element spaces in physical domain
2.0
1.0
1.5
0.8
0.6
1.0
0.4
0.5
0.2
0.0
0.0
0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.5
0.0
V1 transformation in 2D
Colin Cotter
Compatible FEM
0.5
1.0
1.5
2.0
Building finite element spaces in physical domain
X1
GK
(0, 1)
x
X
(0, 0)
(1, 0) X0
Pullbacks to reference element (3D):
▸ ψ ∈ W0 (K ): ψ ○ GK = ψ̂, for ψ̂ ∈ W0 (K̂ ).
▸ ω ∈ W1 (K ): ω ○ GK = J −T ω̂, for ω̂ ∈ W1 (K̂ ).
▸ u ∈ W2 (K ): u ○ GK = J û/ det J, for û ∈ W2 (K̂ ).
▸ ρ ∈ W3 (K ): ρ ○ GK = ρ̂/ det J, for ρ̂ ∈ W3 (K̂ ).
Colin Cotter
Compatible FEM
Terrain-following meshes
▸
All vectors can be evaluated in Cartesian coordinates.
▸
Vector fields in the vertical space Vv1 are aligned with ẑ.
▸
▸
Vector fields in the horizontal space Vh1 are not necessarily
normal to ẑ.
Hence we cannot avoid pressure gradient errors, just as for
C-grid staggered finite difference method.
Colin Cotter
Compatible FEM
Where to store temperature?
The compatible FEM version of the Charney-Phillips staggering
requires to use the same DOFs for temperature as the vertical part
of velocity.
Colin Cotter
Compatible FEM
No spurious hydrostatic modes
0 = − ∂p
∂z + b becomes
∇
⋅
wp dx + ∫Ω w ⋅ ẑb dx = 0, ∀w ∈ Wv2 ,
∫Ω
with p ∈ W3 , b ∈ Wθ2 .
Theorem
With no slip boundary conditions, and a columnar mesh, there is a
one-to-one mapping between b and p (up to a constant). The
mapping is unique if there is a (weak) Dirichlet boundary condition
for p on the top surface.
Colin Cotter
Compatible FEM
Approximation theory
▸
▸
Classical approximation theory relies upon (Pk (K ))m ⊂ Vj (K )
for k + 1-order approximations.
Arnold, Boffi, and Bonizzoni Numer. Math. (2015) showed
that acheivable k is reduced for j > 0 with non-affine
transformations.
Is this a problem for terrain-following columnar meshes and/or on
the sphere?
Colin Cotter
Compatible FEM
Approximation theory
Proposition (Natale and Cotter)
Consider an (n + m)-dimensional element K̂ = T̂ × Ŝ obtained as
tensor product of an n-dimensional simplex T̂ and an
m-dimensional simplex Ŝ. Let FK be a multilinear map, affine
invariant on Ŝ, ν = min(k, n) and En(r +ν),m(r +k) Λk (K̂ ) ⊂ Vk (K̂ ).
Then Pr Λk (K ) ⊂ Vk (K ).
Bottom line
Pullbacks do reduce size of contained polynomial space for k > 0
even for columnar meshes with vertical side-walls.
Colin Cotter
Compatible FEM
Approximation theory
▸
▸
Holst and Stern, FoCM (2012) showed that approximation
order is not reduced provided that mesh is obtained from
global transformation from an affine mesh.
The mapping G ∶ Ω → R4 given by
x = (x, y , z) ↦ G (x) = (
x y z
, , , ∣x∣) ,
∣x∣ ∣x∣ ∣x∣
defines a domain that can be meshed using affine elements.
▸
For terrain-following meshes, C can be composed with the
global mapping to the spherical annulus.
Colin Cotter
Compatible FEM
Approximation theory
Theorem (Natale and Cotter)
Assume that Pr Λk (K̃ ) ⊂ Vk (K̃ ) and let G ∶ Ω̃ → Ω be a C 0
diffeomorphism. Moreover, assume that for any K̃ ∈ Th̃ the
restriction GK ∶ K̃ → K is a C r +2 diffeomorphism, such that
max ∣GK ∣W s
1≤s≤r +2
∞ (Ω̃)
≤ M,
∣GK−1 ∣W 1 (Ω̃) ≤ M
∞
for M > 0. Then, for (n − k)/2 < s ≤ r + 1,
inf
p∈Λkh (Ω)
Colin Cotter
Compatible FEM
∥u − p∥L2 Λk (Ω) ≤ Chs ∥u∥H s Λk (Ω) ,
u ∈ H s Λk (Ω).
Approximation theory
100
k =3
k =2
k =1
k =0
10-1
h1
L2
Error
h2
10-2
10-3
10-1
Colin Cotter
Compatible FEM
h
100
Energy conserving 3D formulation
1
ut + ω × u = −cp θ∇Π − ∇ ( ∣u∣2 + Φ) ,
2
ρt + ∇ ⋅ (ρu) = 0,
θt + u ⋅ ∇θ = 0,
ω = ∇ × u + 2Ω,
where Π is a function of ρ and θ satisfying
∂Π
R
=
Π,
∂θ cv θ
Colin Cotter
Compatible FEM
∂Π
R
=
Π.
∂ρ cv ρ
Energy conserving 3D formulation
▸
u ∈ W02 = {u ∈ W2 ∶ u ⋅ n = 0, on ∂Ω}.
▸
ρ ∈ W3 ,
▸
θ ∈ Wθ ,
▸
Evaluate Π as a pointwise function of ρ and θ.
We define the total vorticity ω ∈ W01 through the weak
approximation
⟨Σ, ω⟩ = ⟨∇ × Σ, u⟩ + ⟨Σ, 2Ω⟩ ,
∀Σ ∈ W01 ,
where W01 = {ω ∈ W1 ∶ ω × n = 0 on ∂Ω} .
This is the projection of ω onto W01 .
Colin Cotter
Compatible FEM
Energy conserving 3D formulation
1
F
⟩ − ⟨∇ ⋅ w, ∣u∣2 + cp θΠ + Φ⟩
ρ
2
w
w
+ ⟨∇h ⋅ ( Pθ (ρcp Π)) , θ⟩ − ⟪[[ Pθ (ρcp Π)]] , {θ}⟫ = 0,
ρ
ρ
⟨φ, ρt + ∇ ⋅ F⟩ = 0,
F
F
⟨γ, θt ⟩ − ⟨∇h ⋅ ( γ) , θ⟩ + ⟪[[ γ]] , {θ}⟫ = 0,
ρ
ρ
0
where F ∈ W2 with ⟨w, F⟩ − ⟨w, uρ⟩ = 0,
⟨w, ut ⟩ + ⟨w, ω ×
∀w ∈ W02 ,
∀φ ∈ W3 ,
∀γ ∈ Vθ ,
∀w ∈ W02 ,
and Pθ (α) ∈ Wθ with ⟨γ, Pθ (α)⟩ − ⟨γ, α⟩ = 0 ∀γ ∈ Vθ .
Colin Cotter
Compatible FEM
E [u, ρ, θ] =
1
2
∫ ρ∣u∣ + cv ρθΠ + ρΦ dx.
2 Ω
We get Ė = ⟨F, ut ⟩ + ⟨cp ρΠ, θt ⟩ + ⟨cp θΠ, ρt ⟩ ,
F
1
= −⟨F, ω × ⟩ + ⟨∇ ⋅ F, ∣u∣2 + cp θΠ + Φ⟩
ρ
2
F
F
− ⟨∇h ⋅ ( Pθ (ρcp Π)) , θ⟩ + ⟪[[ Pθ (ρcp Π)]] , {θ}⟫
ρ
ρ
F
F
+ ⟨∇h ⋅ ( Pθ (cp ρΠ)) , θ⟩ − ⟪[[ Pθ (cp ρΠ)]] , {θ}⟫
ρ
ρ
1 2
− ⟨ ∣u∣ + cp θΠ + Φ, ∇ ⋅ F⟩ = 0.
2
Colin Cotter
Compatible FEM
Summary and outlook
▸
Compatible finite elements provide a way to cleanly separate
divergence-free and curl-free vector fields.
▸
This leads to numerous stability and conservation properties,
including GungHo design requirements.
▸
Tensor-product elements provide an extension to columnar
meshes of prisms or hexahedra.
▸
FEM version of Charney-Phillips staggering is to use the
vertical part of the velocity space to store temperature.
▸
Optimal approximation properties are recovered provided that
a global transformation from an affine mesh is used.
▸
Energy conservation can be extended to 3D compressible too.
Colin Cotter
Compatible FEM
What’s next
▸
We are exploring incorporation of stable advection schemes for
u, θ and ρ into this framework to obtain a discretisation that
does not require explicit (hyper-)viscosity/diffusion for
stability.
▸
See Jemma Shipton’s talk for description of these schemes
and testing in a vertical slice configuration.
For references please see my website and the arXiv.
Colin Cotter
Compatible FEM
Future Weather/Climate Atmospheric Dynamic Core
Problems with lat-lon coordinate for global models
Staggered finite difference methods
!
Revised Monday, November 30, 2009!
• Pole singularities require special filtering
16
• Polar filters do not scale well on massively parallel computers
“overset grids.” There •have
alsoanisotropic
been recent
to use
grids based on octahedrons (e.g.,
Highly
gridattempts
cells at high
latitudes
McGregor, 1996; Purser and Rancic, 1998). A “Fibonacci grid” has also been suggested
(Swinbank and Purser, 2006).
We sought a staggered FD scheme on pseudouniform grids:
Consideration of alternative spatial discretizations:
Lat-Lon
Icosahedral-triangles
Icosahedral-hexagons
Cubed Sphere
Yin-Yang
What goes wrong: Priority Requirements: ••
•
▸ On triangular
•
grids:
spurious
inertia-gravity wave modes.
MPAS
Grids based on icosahedra
• offer an attractive framework for simulation of the global
circulation of the atmosphere; their advantages include almost uniform and quasi-isotropic
▸ On quadratilateral
nonorthogonal
grid,
loss of
resolution over grids:
the sphere. Suchneed
grids are termed
“geodesic,” because they resemble
the geodesic
domes designed by Buckminster Fuller. Williamson (1968) and Sadourny (1968) simultaneously
a new approach
to more homogeneously
discretize geostrophic
the sphere. They constructed grids
consistency inintroduced
Coriolis
term
(if
steady
mode
using spherical triangles which are equilateral and nearly equal in area. Because the grid points
are not regularly spaced and do not lie in orthogonal rows and columns, alternative finiteproperty required).
difference schemes are used to discretize the equations. Initial tests using the grid proved
encouraging, and further studies were carried out. These were reported by Sadourny et al. (1968),
▸ On dual-icosahedral
(hex/pent)
grids:(1970),loss
of (1986).
consistency in
Sadourny and Morel
(1969), Sadourny (1969), Williamson
and Masuda
The grids are constructed from an icosahedron (20 faces and 12 vertices), which is one of
Coriolis termthe(if
steady
geostrophic
mode
required).
five Platonic
solids. A conceptually
simple scheme for
constructing property
a spherical geodesic grid
Efficient on existing and proposed supercomputer architectures
Fig. 11.8: Various ways of discretizing the sphere. Figure made by Bill Skamarock of NCAR.
Scales well on massively parallel computers
Well suited for cloud (nonhydrostatic) to global scales
Capability for local grid refinement and regional domains
Conserves at least mass and scalar quantities
Mesoscale & Microscale Meteorology Division / ESSL / NCAR
is to divide the edges of the icosahedral faces into equal lengths, create new smaller equilateral
non-orthogonal: Thuburn
J.SeeSci.
Comp.
(2012)
triangles in the and
plane, andCJC.
then projectSIAM
onto the sphere.
Fig. 11.9.
One can construct
a more
homogeneous grid by partitioning the spherical equilateral triangles instead. Williamson (1968)
Sadourny (1968) use
slightly different
techniques
to construct
their grids.
However, both
loss of consistency: and
Thuburn,
CJC,
and
Dubos.
GMD
(2014)
begin by partitioning the spherical icosahedral triangle. On these geodesic grids, all but twelve of
Colin Cotter
Compatible FEM
the cells are hexagons. The remaining twelve are pentagons. They are associated with the twelve
vertices of the original icosahedron.
Williamson (1968) chose the nondivergent shallow water equations to test the new grid.
He solved the two-dimensional nondivergent vorticity equation
Velocity advection
▸
We’re currently using a vertical slice version of the
vorticity-based advection scheme given in McRae and Cotter,
QJRMS (2014).
▸
A vorticity is diagnosed, advanced using a stable advection
scheme, and then the velocity is updated in a consistent way.
▸
Some concerns at the Met Office over using vorticity, so we
are developing stable advection schemes for velocity directly.
Colin Cotter
Compatible FEM
Helmholtz solver
Analytic elimination of ∆b leads to
∆t
′
∫ w ⋅ Σ∆u dx −
∫ ∇ ⋅ w∆p dx = −Ru [w],
2 Ω
Ω
2 ∆t
∇ ⋅ ∆u) dx = −Rp [φ],
∫ φ (∆p + c
2
Ω
▸
▸
▸
▸
∀w ∈ V2 ,
∀φ ∈ V3 .
Usual approach of eliminating u to give Helmholtz problem for
p isn’t practical as inverse V2 mass matrix is dense.
We precondition GMRES by approximate Schur complement
using diagonal of the V2 mass matrix.
Schur complement itself is approximated by GAMG.
Hybridisation would reduce the required number of GAMG
calls per iteration.
Colin Cotter
Compatible FEM
Hybridisation
Replace W2 with W̄2 , which has the same basis functions but
without inter-element continuity. Enforce inter-element continuity
with Lagrange multipliers integrated over element edges.
Seek u ∈ W̄2 , φ ∈ W3 , Λ ∈ T (W2 ) such that:
∫ w ⋅ u − ∇ ⋅ wφ dx + ∫ [[w ⋅ n]]λ ds = Ru [w],
Ω
Γ
∫ c0 γφ + γ∇ ⋅ u dx = Rφ [γ],
Ω
∫ µ[[u ⋅ n]] ds = 0,
Γ
Colin Cotter
Compatible FEM
∀w ∈ W̄2
∀γ ∈ W3
∀µ ∈ T (W2 ).
Matrix equations:
M̄u û + C̄ T φ̂ + LT λ̂ = R̄u ,
c0 Mφ φ̂ − C̄ û = R̂φ ,
Lû = 0.
Becomes
M̄
C̄ T
(L 0) ( u
)
C̄ c0 Mφ
▸
▸
▸
▸
−1
LT
( ) λ̂ = RHSλ .
0
Symmetric (even if include Coriolis) positive-definite matrix,
same conditioning as Helmholtz operator.
Same element stencil as if eliminated φ.
Not singular (modulo boundary conditions) as c0 → 0.
Multigrid algorithms exist (Gopalakrishnan and Tan (2009)).
Colin Cotter
Compatible FEM
Construction of S
ge
ge ∶ e ′ → e, x = ge (x′ ).
Definition (Piola transformation)
The Piola transformation u′ ↦ u:
u ○ ge =
Colin Cotter
Compatible FEM
1
det
∂ge
∂x′
∂ge ′
u
∂x′
u ⋅ n dx = g∗e (u ⋅ n dx)
′
′
′