hermitian compact operators on the cubed-sphere -...

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HERMITIAN COMPACT OPERATORS ONTHE CUBED-SPHERE

Jean-Pierre Croisille

Departement de mathematiques

Universite de Lorraine - Metz, France

National Center for Atmospheric Research

April 27, 2012

Jean-Pierre CROISILLE - Universite de Lorraine - Metz, France NCAR - April 27, 2012

Outline

1 The Cubed-Sphere Grid

2 Three-point Hermitian Derivative in one-dimension

3 Hermitian derivatives on great circles

4 Approximate high-order spherical gradient

5 The cosine-bell advection test-case

6 A high-order eigenvalue problem on the sphere.


The Cubed-Sphere grid(1)

Definition

The Cubed-Sphere is composed of six square-like patches which are the centralprojection onto the sphere of the six faces of the enscribed cube.

The grid is regular in the equiangular coordinates (ξ, η).

Figure: The cubed-sphere grid with N = 16 (left) and N = 32 (right).



Figure: The cubed-sphere grid with N = 8.



Figure: The cubed-sphere grid with N = 16.



Topology of the cubed-sphere

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

F(I) E(II) B(III)W(IV)

N(V)

S(VI)

NOTATION FOR THE 6 PATCHES OF THE CUBED−SPHERE

Figure: The 6 squared patches of the cubed-sphere grid. The number (I), (II),(III), (IV ), (V ), (V I) corresponds to the face Front (F), East(E), Back (B), West (W),North (N) and South (S), respectively.


Coordinate systems on the cubed-sphere grid

Equiangular coordinates

The angle ξ along the equator and η along the central meridian of each patch area local system of coordinates.

Six coordinate systems:

(ξI , ηI), (ξII , ηII), (ξIII , ηIII), (ξIV , ηIV ), (ξV , ηV ), (ξV I , ηV I).

(1)

Gnomonic coordinates

The values X = tan ξ and Y = tan η are useful in the computations. They are calledthe gnomonic coordinates.

Cartesian coordinates

The reference cartesian frame is (0, i, j, k). The directions of the unit vectors i, j, k

are associated to the Front, East and North directions, respectively.


Grid functions on the cubed-sphere

Discretization

Our main object is a grid function on the cubed-sphere. The grid is cartesian in the(ξ, η) coordinate systems. The full gridding is therefore made of six identical squaregrids.

Grid function

A cubed-sphere grid function is given by the values

uki,j , −M ≤ i, j ≤ M, k = I, II, III, IV, V, V I (2)

with M = N/2.

Size

If N is the size of a patch there are 6(N + 1)2 − 12(N − 1)− 2× 8 = 6N2 + 2 pointson the cubed-sphere.


Metric of the cubed-sphere

Covariant base

gξ =∂x

∂ξ, gη =

∂x

∂η(3)

The metric tensor is

G =

»gξ · gξ gξ · gη

gη · gξ gη · gη

–(4)

Metric tensor

The metric tensor can be expressed in function of the gnomonic coordinates as

G =r2

δ4(1 + X2)(1 + Y 2)

»1 + X2 −XY

−XY 1 + Y 2

–, (5)

where δ =√

1 + X2 + Y 2.


Metric of the cubed-sphere (2)

Contravariant base

The contravariant vectors are deduced bygξ = G11gξ + G12gη ,

gη = G21gξ + G22gη ,(6)

where

G−1 =

»G11 G12

G21 G22.

–(7)

Explicit form

Example: the contravariant base (gξ, gη) is given for the patch Front by

gξ =1

x(1 + X2)

24 −X

1

0

35 , gη =1

x(1 + Y 2)

24 −Y

1

0

35 (8)


Three-point Hermitian Derivative Operator

Finite Difference form

To a given a grid function (ui)i∈Z we associate the “Hermitian discrete derivative” by

σxux,i = δxui, i ∈ Z, (9)

where σx, δx are the finite difference operators

σxui =1

6ui−1 +

2

3ui +

1

6ui+1, δxui =

ui+1 − ui−1

2h. (10)

Properties

Non local finite difference operator

[ux] = (σx)−1δx[u]. (11)

Fourth-order accurate at each point

ux,i = u′(xi) + O(h4) (12)


Hermitian Derivative Operator on an irregular periodicgrid

Principle

Irregular periodic grid given by xj = ϕ(tj) 0 ≤ j ≤ N − 1 with tN = t0. The gridtj = j∆t is an equispaced discretisation of [0, 1] with 1− periodicity.

The chain rule at point tj is written as

u′(ϕ(t)) =(u ◦ ϕ)′(t)

ϕ′(t), 0 < t < 1. (13)

The Hermitian derivative at xj is

ux,j =(u ◦ ϕ)t,j

ϕt,j, 0 ≤ j ≤ N − 1, (14)

where ϕt,j (resp. (u ◦ ϕ)t,j ) is the hermitian derivative of ϕ(t) (resp. (u ◦ ϕ)(t))

at point tj .


Coordinate great circles

Interest of the Cubed-Sphere

The coordinate lines are sections of great circles. This suggests to performone-dimensional interpolations along these circles using 2π−periodicity.

A “coordinate” great circle

−pi/4 0 pi/4−\pi/4

0

\pi/4

−π/4 ≤ ξF ≤ π/4

−π/

4 ≤

η F ≤

π/4

FRONT PATCH

η=η0

−pi/4 0 pi/4−pi/4

0

pi/4

−π/4 ≤ ξE ≤ π/4

−π/

4 ≤

η E ≤

π/4

EAST PATCH

−\pi/4 0 \pi/4−\pi/4

0

\pi/4

−π/4 ≤ ξB ≤ π/4

−π/

4 ≤

η B ≤

π/4

BACK PATCH

−\pi/4 0 \pi/4−\pi/4

0

\pi/4

−π/4 ≤ ξW

≤ π/4

−π/

4 ≤

η W ≤

π/4

WEST PATCH

Figure: Representation of a coordinate great circle. It coincides with the coordinateline η = η0 of patch Front (top left). Then it crosses the patch East (top right) along aset of points (ξE

i , ηEi ). Then (bottom left) on patch Back it coincides with the

coordinate line η = π − η0. Finally (bottom right) it crosses the patch West.


Networks of great circles on the cubed-sphere

Six networks of great circles

System (Iα): based on ξ coordinate lines of face I,

System (Iβ): based on η coordinate lines of face I,

System (IIα): based on ξ coordinate lines of face II,

System (IIβ): based on η coordinate lines of face II,

System (Vα): based on ξ coordinate lines of face V,

System (Vβ): based on η coordinate lines of face V.


Networks of great circles on the cubed-sphere

Figure: The network (Iα) of great circles. This network corresponds to the iso η

coordinate lines of patch F . The great circles are plotted with o symbols. Left: view ofthe Front (I) patch. Right: view of the East patch. The displayed grid corresponds toN = 16 with 1538 points.


Computing the spherical gradient

Step 1: Grid points along a great circle

Consider a great circle of network Iα. It coincides with an iso −η coordinate line of thepatch Front ηI = ηI

0 . The angle α is the curvilinear absissa along this coordinate linewith eastward orientation. On the patch Front, the angle α is such that

−α0(η) < α < α0(η), α0(η) = atan

√2 tan η

2

!(15)

For a spherical grid function uki,j , −M ≤ i, j ≤ M , I ≤ k ≤ V I, the data are the 4N

values u(αl), 0 ≤ αl ≤ 2π.

Face I: u is given by the data uIi,j0

−M ≤ i ≤ M .

Face II: The points are defined as the intersections of the great circle with thecoordinate lines ξ = ξII

i . This interestion system defines the set of angles anglesαl for the face II.

Face III: Again the great circle coincides with a coordinate line of face III. Weadopt the data uIII

i,−j0−M ≤ i ≤ M .

Face IV: As for face II the points are defined by an interection system betweenthe great circle and the coordinate lines ξ = ξIV

i .


Computing the spherical gradient (2)

Step 2: Assembling values along a great circle

On faces II (East) and IV (West), the values at the points of the great circle are given byan interpolation. Along each “vertical” coordinate lines ξ = ξII

i , −M ≤ i ≤ M , weadopt a cubic spline interpolation. End values are specified (for example) by the “not aknot” condition.



Values along a great circle

Figure: Values on a great circle of the network (Iα).



Step3: Hermitian derivatives along a great circle

Using discrete one-dimensional periodic data along the great circle, allows to evaluatethe hermitian gradient. This gives a “fourth order” accurate approximation of thederivative along the circle.

Hermitian derivatives for six networks of great circles

As a result we obtain on each patch I, II, III, IV, V, V I approximations of thederivatives 8>><>>:

∂u(α, η)

∂α |η,

∂u(ξ, β)

∂β |ξ

(16)


Computing the spherical gradient(5)

Step 4: Evaluating the spherical gradient

The gradient ∇su is obtained using the formula of differential geometry

∇su =∂u

∂ξ |ηgξ +

∂u

∂η |ξgη . (17)

where (gξ, gη) is the contravariant basis at the point (ξ, η).

Change of variables

The function u(ξ, η) is expressed using the change of variables (ξ, η) 7→ α(ξ, η) as

u(ξ, η) = u(α(ξ, η), η) = u(ξ, β(ξ, η)). (18)

The chain rule gives 8>><>>:∂u

∂ξ |η=

∂u

∂α |η

∂α

∂ξ |η,

∂u

∂η |ξ=

∂u

∂β |ξ

∂β

∂η |ξ.

(19)


Computing the spherical gradient(6)

Spherical gradient

The spherical gradient at each point of the Front patch is deduced from ∂u∂α |η and ∂u

∂β |ξby

∇su =∂u

∂α |η

„cos η

1 + tan2 ξ

1 + cos2 η tan2 ξ

«gξ +

∂u

∂β |ξ

„cos ξ

1 + tan2 η

1 + cos2 ξ tan2 η

«gη . (20)

Spherical gradient approximation

Approximating the derivatives ∂u∂α |η and ∂u

∂β |ξby their hermitian approximations gives

the final formula (on patch Front):

∇s,huIi,j = uI

α,i,j

„cos ηj

1 + tan2 ξi

1 + cos2 ηj tan2 ξi

«gξ

i,j

+ uIβ,i,j

„cos ξi

1 + tan2 ηj

1 + cos2 ξi tan2 ηj

«gη

i,j

(21)

Final approximation on the cubed-sphere

Analog expressions hold for the five other patches.


Accuracy of the calculated gradient

Test 1

We report in Table 1 the error obtained for the gradient of the oscillating function

u(x, y, z) = sin(10πx) sin(2πy) sin(6πz), (22)

restricted to the unit sphere.

N=8 rate N=16 rate N=32 rate N=64 rate N=128

e∞ 23.475 2.99 2.949 4.61 0.121 4.07 7.205(-3) 3.91 4.774(-4)∆ξ in km 1250 625 312.5 156.25 78.12

Nb of grid points 386 1538 6146 24578 93306

Table: Convergence rate of the hermitian gradient of the oscillating functionu(x, y, z) = sin(10πx) sin(2πy) sin(6πz) restricted to the unit sphere.


Accuracy of the calculated gradient (2)

Test 2

We consider the function the gradient of the functionu(x, y, z) = exp(x) + exp(y) + exp(z) restricted to thye unit sphere.


e∞ 5.323(-4) 4.07 1.912(-5) 4.00 1.191(-6) 4.00 7.432(-6) 3.72 5.622(-9)∆ξ in km 1250 625 312.5 156.25 78.12

Nb of grid points 386 1538 6146 24578 93306

Table: Convergence rate of the hermitian gradient of the functionu(x, y, z) = exp(x) + exp(y) + exp(z) restricted to the unit sphere.


Cosine-bell test case (D. Williamson et al.)

Advection equation on the sphere

Test problem : propagation of a cosine-bell at constant solide spherical velocityaround the earth.

Serves as a preliminary test to evaluate the accuracy of numerical methods forthe shallow water system

Reference results widely reported in the literature by various kinds of spatialapproximations: spectral, FV MUSCL, SE, DG, etc, on various grids:longitude/latitude, icosahedral, cubed-sphere, etc.

Setting of the problem

h(x, t)=height of the bell. ∂th(x, t) + c · ∇sh(x, t) = 0

h(x, 0) = h0(x)(23)

The velocity is a “constant” solid velocity.


The Cosine-Bell at initial position

−5

0

5

x 106

−6−4

−20

24

6

x 106

−6

−4

−2

0

2

4

6

x 106

F

W

S

N

E

B

0

100

200

300

400

500

600

700

800

900

1000

Figure: Initial position of the cosine-bell. Cubed-sphere grid with N = 64.


Spatial approximation

Centered finite-difference scheme

Purpose: assess the accuracy of the new approximate gradient. We thus use thecentered semi-discrete fourth-order scheme.

dhki,j(t)

dt+ c ·∇s,hhk

i,j(t) = 0, −M ≤ i, j ≤ M, I ≤ k ≤ V I. (24)

Filtering

No upwinding is necessary to handle a linear wave propagation problem (Maxwellequations, aeroacoustic problems, etc.). A simple high-frequency filter is enough tosupress high-frequency dispersive oscillations of a centered scheme.

Tenth-order filter

Suppression of the dispersive effects supported by by the “1/− 1” mode. Such a filteris common in aeroacoustics computations.


Filtering the 1/− 1 mode

Filtering of a grid function

For a gridfunction (uj)j∈Z the filter is the linear operator (uj)j∈Z 7→ (uF,j)j∈Z definedby

uF,j =X

k

fkuj+k, (25)

where the coefficients fk are given by0BBBBBBB@

f0

f1 = f−1

f2 = f−2

f3 = f−3

f4 = f−4

f5 = f−5

1CCCCCCCA=

0BBBBBBB@

772/1024

210/1024

−120/1024

45/1024

−10/1024

1/1024

1CCCCCCCA. (26)

This is an interpolation operator of order 10.


Fourth-order time-stepping scheme

Runge-Kutta RK4 scheme

For the time-dependent system

d

dtV (t) = AV (t), J ∈ MN (R), (27)

the standard explicit RK4 scheme is8>>>><>>>>:k0 = AV n

k1 = A(V n + 12∆t k0)

k2 = A(V n + 12∆t k1)

k3 = A(V n + ∆t k2)

V n+1 = V n + ∆t`

16k0 + 1

3k1 + 1

3k2 + 1

6k3

´.

(28)


Cosine-bell test case: numerical results (1)

History of the errors

0 2 4 6 8 10 120

0.01

0.02

0.03

0.04

0.05

0.06

time(days)

rela

tive

erro

r

L1 errorL2 errormax error

Figure: History of the three relative errors L1, L2, L∞ for the cosine-bell advectionproblem for a cubed-sphere grid of size N = 40.



CFL Direction L1 error L2 error L∞ error Maximum Minimum

1.0 α = 0deg 5.43601(-2) 3.26394(-2) 2.65404(-2) 2.50503(-2) -2.65404(-2)α = 45deg 5.11205(-2) 2.93375(-2) 2.24284(-2) 2.11063(-2) -2.24284(-2)

0.5 α = 0deg 4.0020(-2) 2.25259(-2) 1.91928(-2) 2.87524(-2) 1.91193(-2)α = 45deg 3.45757(-2) 1.86116(-2) 1.43016(-2) 1.27016(-2) -1.43016(-2)

Table: Relative error for the cosine-bell advection problem with the fourth-ordercentered hermitian scheme (24) combined with the RK4 time-stepping scheme. Themesh size is 40× 40× 6 after 12 days.



Isolines

0

0

0

0

00

0

00

00

0

0

0 0

0

0

00

0

0

0

0

800

800

Cosine bell problem: calculated solution

−30 −20 −10 0 10 20 30−30

−20

−10

0

10

20

30

−20−10

−10

−10

−10

−10−10

−10

0

0

0

0

0

0

00

0

0

0

0

0

0

0

0

0

0

0

0

0 0

0

0

0

0

0

0

00

0

00

00

10

10

1010

10

10

10

2020

Cosine bell problem: max error

−30 −20 −10 0 10 20 30

−30

−20

−10

0

10

20

30

Figure: Contours after one rotation with α = π/4. Left: numerical solution.Right: error. The cubed-sphere uses N = 40. The propagation is orientedtowards north-east.


Compact Laplacian on the cubed-sphere

Spherical Laplacian

Spherical Laplacian:∆su = ∇s · (∇su), (29)

where ∇s· and ∇s are the spherical divergence and the spherical gradient.

Coordinate expression of the Laplacian

In the coordinate system (ξ, η) the spherical divergence operator of the vector field F

is expressed as

∇ · F =1√

G

„∂

∂ξ(p

GF · gξ) +∂

∂η(p

GF · gη)

«. (30)

where G isG =

p| det G|. (31)


Laplacian on the cubed-sphere (2)

Compact Laplacian on each patch

From the gradient ∇su, we deduce on each patch the grid functions u1(ξ, η) andu2(ξ, η),

u1(ξ, η) =p

G∇su · gξ, u2(ξ, η) =p

G∇su · gη (32)

Using the discrete gradient above an approximation of the two functions u1 and u2 oneach patch is

(u1,h)ki,j =

pG“∇s,huk

i,j

”· gξ

i,j , (u2,h)ki,j =

pG“∇s,huk

i,j

”· gη

i,j . (33)

for −M ≤ i, j ≤ M and I ≤ k ≤ V I.



Boundary conditions

Here there is no use of periodicity around great circles. Instead the discrete hermitianLaplacian is coordinate dependent. Thus we need boundary conditions along the fouredges of each patch to define local hermitian derivatives. We adopt the one-sidedapproximate derivative

u′(x) = δ+x uj −

1

2(δ+

x )2uj +1

3(δ+

x )3uj −1

4(δ+

x )4uj + O(h4), (34)

where the forward difference operator δ+x is

δ+x uj =

uj+1 − uj

h. (35)

The fourth-order approximate derivative is

ux,i =1

12h(−25ui + 48ui+1 − 36ui+2 + 16ui+3 − 3ui+4) (36)



Discrete hermitian Laplacian

The discrete Laplacian is

∆huki,j =

1√

G

“(u1,h)k

ξ,i,j + (u2,h)kη,i,j

”. (37)

This is a formula dependent on the coordinate system (ξ, η) on each patch.


Numerical test for the Laplacian

Spherical harmonics

The spherical harmonic with index (n, m), −n ≤ m ≤ n, 0 ≤ n is defined in sphericalcoordinates (θ, λ) by

fmn (x) = P

|m|n (sin θ)eimλ. (38)

The function P|m|n (z) is the associated Legendre polynomial of order n, m with

standard normalization given by Z 1

−1P|m|n (x)2dx = 1. (39)

Main property: the spherical harmonics are the eigenfunctions of the sphericalLaplacian.


Numerical test for the Laplacian (2)

Eigenfunction numerical test

The eigenvalue associated with fmn (x) is

λn = −n(n + 1). (40)

The test of accuracy of the hermitian Laplacian is

e(fnm) = max

−M≤i,j≤M,I≤k≤V I|`∆h,sfn

m − λnfnm

´|, (41)


Numerical results


e(f11 ) 8.6508(-3) 3.74 6.4381(-4) 4.18 3.5512(-5) 4.34 1.7496(-6) 3.35 1.7066(-7)

e(f23 ) 2.3477(-1) 3.70 1.8073(-2) 4.06 1.0805(-3) 3.68 8.3954(-5) 3.51 7.3302(-6)

e(f48 ) 1.9702(1) 3.54 1.6991(0) 3.24 1.7884(-1) 3.98 1.1325(-2) 4.00 7.0537(-4)

e(f914) 2.0246(2) 2.92 2.666(1) 3.06 3.1909 4.00 1.9809(-1) 3.96 1.2696(-2)

∆ξmax in km 1250 625 312.5 156.25 78.12Nb of grid points 386 1538 6146 24578 93306

Table: Convergence rate of the Laplacian of several spherical harmonics functions.The reported values are given in (41) .


Conclusion

Interest of the great circles approach

Uniformly high-order treatment of the sphere.

Avoids complex boundary treatment (ghost points, etc) at the inter patchboundaries.

High-order accuracy relies on purely one-dimensional periodic computations.

Only a (large) series of one-dimensional periodic “tridiagonal” systems to solve.

Implementation so far

Code in matlab so far.

Fourth-order Hermitian gradient is implemented.

Simple test-cases (linear advection) are tractable for N = 40.


Ongoing work

Compact hermitian operators

Intrinsic design of higher-order differential operators such as spherical Laplacian,divergence or Biharmonic using great circles methodology.

No inter-patch boundary treatment depending of the differential operators to beapproximated.

Higher-order accuracy easy to obtain using very high-order hermitian derivatives(6-th order, 8-th order or higher.)

Implementation of the Laplace Tidal Equations.

MUSCL finite-volume schemes

High-order gradients are important for a stable, accurate MUSCL finite-volumemethod.

Various settings for the MUSCL are possible: cell-center or vertex-center.


Some perspectives

Models

Investigation of complex linear wave problems on the sphere (atmosphere andocean).

Numerical computation of eigenmodes of linear operators of interest inclimatology/oceanography.

Scientific computing

Essential issue in climatology: local grid refinement on the cubed-sphere.

Parallel implementation of the great circles approach seems attractive.

Numerical analysis

Prove some convergence results of the great circles approach to compute thespherical gradient.


hermitian compact operators on the cubed-sphere -...

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