(High-order) nite elements for the shallow-water
equations on the cubed sphere
E.Kritsikis, T. Dubos
University of Paris 13 & LMD Ecole polytechnique
PDEs on the Sphere 2014
Plan
1
Formulation
2
Fixing the spectral gap
3
Sphering the cube
4
Results
2/18
The problem
Shallow water
∂t h + ∇ · U = 0 ,
∂t u + qU + ∇B = 0 ,
⊥
U = hu
B = hg + u 2 /2
Questions
ensure the conservation requirements (mass, energy, momentum, PV)
get good dispersion properties
do it on the sphere
Formulation
3/18
The Hamiltonian structure
R
H (u , h) = 12 h(gh + u 2 )
The Poisson bracket
dF
dt
= {F , H } =
Z
q·
∂F
∂H
×
∂u
∂u
+
∂H
∂F
∂F
∂H
div
−
div
∂h
∂u
∂h
∂u
track evolution of any F (h, u )
recover eqs of motion if U := ∂ H /∂ u , B := ∂ H /∂ h
antisymmetry → energy conservation
Procedure
dene (U , B ) on U 0 × H0 i.e. as functional derivatives of H
(δ u , U ) + (δ h, B ) = δ H
dene antisymmetric bilinear form J on U 0 × H0 s.t.
U ˆ , U B
Bˆ
= (q , Û × U ) + (B , div Û ) − (B̂ , div U )
∂t (Û , u ) + ∂t (B̂ , h) = J
Formulation
4/18
Spaces (Cotter & Shipton '13)
u, U ∈ U
h, B ∈ H
lose regularity going left to right
Z o
∇⊥ /
rot
U o
div /
H
∇
Formulations
Solve ∀û ∈ U, (U , û ) = (hu , û )
Compute ∂t h = − div U
Compute (for discr.) ∀ĥ ∈ H, the (B , ĥ) = (u 2 /2 + hg , ĥ)
Solve ∀q̂ ∈ Z, (q̂h, q ) = (∇⊥ q̂ , u )
Solve ∀Û ∈ U, (Û , ∂t u ) + (Û , qU ⊥ ) − (div Û , B ) = 0
Note : we have (q̂ , ∂t (hq )) = (∇⊥ q̂ , ∂t u ) = (∇⊥ q̂ , qU ⊥ ) = (q̂ , curl(qU ⊥ )),
weak form of the PV transport ∂t (hq ) + div(qU ) = 0.
Formulation
5/18
FE spaces
Triangle mesh + RT
P1
∇⊥ /
RT0
div / P d
0
2N div. vs N rot. DOFs ⇒ numerical modes
(Gaÿmann '12)
Quad mesh + ⊗
A
Pn
∂x /
B
P n −1
A⊗A
∇⊥ /
Fixing the spectral gap
(?)
div / B ⊗ B
A ⊗ B~ι + B ⊗ A~j
6/18
Linear analysis for
P2 − P1d
Regular periodic 1D, (h, u , v ) ∼ exp i (kx − ω t )
ω = 0 : transport of PV ⇒ geostrophic balance
ω 2 = f 2 + gH k 2 : 2 DOFs per element breaks translational invariance ⇒
spectral gap
c=1 f=10
40
35
DG1
continuous
ω
30
25
20
15
10
0
10
20
N
30
40
(Melvin '13)
Fixing the spectral gap
7/18
Fixing the spectral gap : the FDn − FVn−1 pair
let B be the image of a reconstruction operator (hk ) 7→ h s.t.
Z x
x
+1
k
h ( x ) dx = hk
k
e.g. a nth order FV reconstruction with h ∈ Pnd−1
a basis is bk (x ) with
Z x +1
j
x
bk (x ) dx = δjk
h=
−→
j
let A be the span of the
a k (x ) =
Z x
−∞
(bk +1 (x ) − bk (x )) dx
P
h 7→ k h(xk )ak (x ) is a nth order
∂
interpolation and A →
B
Fixing the spectral gap
k
bk (x )
Z x
x
k
+1
h(x ) dx
k
1.5
0
x
X
0
0
a(x)
b(x)
1
0.5
0
−0.5
−2
0
2
4
8/18
Fixing the spectral gap : the FD2 − FV1 pair
c=1 f=10
40
35
DG1
continuous
ω
30
25
20
15
10
0
Fixing the spectral gap
10
20
N
30
40
9/18
Fixing the spectral gap : the FD2 − FV1 pair
c=1 f=10
40
35
FV1
continuous
ω
30
25
20
15
10
0
10
20
N
30
40
Extends to arbitrarily-high order (but stencil increases)
Fixing the spectral gap
9/18
Wave velocities
Fixing the spectral gap
10/18
Covariant formulation
Patch-based approach
6 squares mapped to the sphere
(x 1 , x 2 ) 7→ r (x 1 , x 2 )
J
Gij = ∂i r · ∂j r
p
= (e1 × e2 ) · r = det Gij
solve for J-weighted height and J-weighted
contravariant velocity components
m=Jh
u ∈ H (div)
↔
1
u = u i ∂i r
J
continuity of u i
copy DOF across patch boundary
Sphering the cube
↔
11/18
Covariant formulation
Hamiltonian
gm Gij j i
m
+ 2u u
2
J
J
δH
gm Gij j i
B=
=
+
uu
δm
J 2J 2
δH
m
U i = i = u i = hu i
δu
J
1
H(u j , m) =
Z
Poisson bracket
Z
δF =
{F, H} = J
J
Sphering the cube
Gij δF j δF
δu +
δm
J δu i δm δF /δ u i
δF /δ m
Uˆ , Uˆ
= εij q Û i U j +
B
Bˆ
i
i
,
Z
δH/δ u i
δH/δ m
B ∂i Û i − B̂ ∂i U i
12/18
The weak and the strong
Z
εij ∂j
/ Ui
∂i /
M,
U 1 = A ⊗ B,
Z = A ⊗ A,
U 2 = B ⊗ A,
M=B⊗B
Gij i
mu ,
Û i , U i ∈ U i
J2
g
1G
m + 2ij u j u i ,
(m̂, B ) = m̂
m̂, B ∈ M
J
2J
Gij j
ψ̂, q ∈ Z
ψ̂ m, q = ψ̂ Jf − εik
u ∂ ψ̂ ,
J k
G
Û i , ij ∂t u j + Û i , εij U j q − B ∂i û i = 0,
Û i , u i ∈ U i
J
∂t m + ∂i U i = 0
Û j ,
Gij i
U
J
=
Û j ,
Strong cont. eq. → can be coupled to FV transport schemes
Sphering the cube
13/18
FD3/FV2 : basis functions
Results
14/18
FD3/FV2 : order of individual operators
0−form
−2
1−form
−2
10
10
L error
L error
∞
∞
4−th order
−4
10
−6
−4
10
10
−8
−5
10
10
−10
10
3rd order
−3
10
−6
1
2
10
curl
−2
10
10
1
10
L∞ error
4th order
−4
10
−6
2nd order
10
10
−8
−4
10
10
−10
−5
1
10
Results
L∞ error
−2
−3
10
10
10
Laplacian
−1
10
2
10
2
10
10
1
10
2
10
15/18
Results : rotating shallow-water
test cases from Williamson et al. (1991)
3rd-order convergence for solid-body rotation, other tests closer to 2nd-order
Results
16/18
Results : Galewsky test-case
Results
17/18
Summary
Discretize Hamiltonian, Poisson bracket
Mimetic properties result from exact suites
Use quadrangles to balance DOFs and avoid numerical modes
Standard higher-order spaces suer from spectral gap
Use 1 DOF per element to restore discrete translational invariance
New 1D spaces based on interpolation-reconstruction operators
Promising results on cubed sphere
Ongoing work : stabilization, eciency
Results
18/18