Slide 1 Numerical Methods: Series-Expansion Methods Slide 1 Agathe Untch e-mail: [email protected] (office 011) Series-Expansion Methods

Numerical Methods: Series-Expansion Methods

Slide 1

Slide 1

Agathe Untch

e-mail: [email protected]

(office 011)

Series-Expansion Methods Series-Expansion Methods


Slide 2

Slide 2

Acknowledgement

This lecture draws heavily on lectures by Mariano Hortal (former Head of the Numerical Aspects Section, ECMWF)

and on the excellent textbook

“Numerical Methods for Wave Equations in Geophysical Fluid Dynamics” by Dale R. Durran (Springer, 1999)

Acknowledgement

This lecture draws heavily on lectures by Mariano Hortal (former Head of the Numerical Aspects Section, ECMWF)

and on the excellent textbook

“Numerical Methods for Wave Equations in Geophysical Fluid Dynamics” by Dale R. Durran (Springer, 1999)


Slide 3

Slide 3

Introduction

The ECMWF model uses series-expansion methods in the horizontal and in the vertical

- It is a spectral model in the horizontal with spherical harmonics expansion with triangular truncation.

- It uses the spectral transform method in the horizontal with fast Fourier transform and Legendre transform.

- The grid in physical space is a linear reduced Gaussian grid.

- In the vertical it has a finite-element scheme with cubic B-spline expansion.

This lecture explains the concept of series-expansion methods (with focus on the spectral method), their numerical implementation, and what the above jargon all means.


Slide 4

Slide 4

Schematic representation of the spectral transform method in the ECMWF model

Grid-point space -semi-Lagrangian advection -physical parametrizations -products of terms

Fourier space

Spectral space -horizontal gradients -semi-implicit calculations -horizontal diffusion

FFT

LT

Inverse FFT

Inverse LT

Fourier space

FFT: Fast Fourier Transform, LT: Legendre Transform


Slide 5

Slide 5

Introduction

Series-expansion methods which are potentially useful in geophysical fluid dynamics are

- the spectral method

- the pseudo-spectral method

- the finite-element method

All these series-expansion methods share a common foundation


Slide 6

Slide 6

Fundamentals of series-expansion methodsFundamentals of series-expansion methods

0)(

fHt

f

)(),( 00 xftxf

(1)

(1a)

We demonstrate the fundamentals of series-expansion methods on the followingproblem for which we seek solutions:

Partial differential equation:(with an operator H involvingonly derivatives in space.)

Initial condition:

Boundary conditions: Solution f has to fulfil some specified conditions on the boundary of the domain S.

To be solved on the spatial domain S subject to specified initial and boundary conditions.


Slide 7

Slide 7

(3)

Fundamentals of series-expansion methods (cont)Fundamentals of series-expansion methods (cont)

1

)()(),(i

ii xtatxf

The basic idea of all series-expansion methods is to write the spatial dependenceof f as a linear combination of known expansion functions )(xi

1)( ii x should span the L2 space, i.e. a HilbertThe set of expansion function space with the inner (or scalar) product of two functions defined as

dxxfxgfgS

)()(, * (4)

The expansion functions should all satisfy the required boundary conditions.


Slide 8

Slide 8


)(),...,(1 tata N

The task of solving (1) has been transformed into the problem of finding the unknown coefficients

in a way that minimises the error in the approximate solution.

N

iii xtatxf

1

)()(),(ˆ

Numerically we can’t handle infinite sums. Limit the sum to a finite numberof expansion terms N

(3a)

f̂ is only an approximation to the true solution f of the equation (1).

=>


Slide 9

Slide 9

Strategies for minimising the residual R: 1.) minimise the l2-norm of R 2.) collocation strategy 3.) Galerkin approximation (Details on the next slide.)

)ˆ()ˆ(ˆ

fRfHt

f

However, the real error in the approximate solution can’t be determined. A more practical way to try and minimise the error is to minimise the residual R instead:

ff ˆ

(5)

Each of these strategies leads to a system of N coupled ordinary differential

equations for the time-dependent coefficients ai(t), i=1,…, N.

0)(

fHt

f(1) <=>


R = amount by which the approximate solution fails to satisfy the governing equation.

f̂


Slide 10

Slide 10

2.) Collocation method Constrain the residual by requiring it to be zero at a discrete set of grid points

2/1*

2/1

2))(ˆ())(ˆ()ˆ(),ˆ()ˆ(

S dxxfRxfRfRfRfR

NiallforxfR i ,...,10))(ˆ(

1.) Minimisation of the l2-norm of the residual Compute the ai(t) such as to minimise

3.) Galerkin approximation

Require R to be orthogonal to each of the expansion functions used in the expansion of f, i.e. the residual depends only on the omitted basis functions.

.,...,10)())(ˆ()ˆ(, * NiallfordxxxfRfRSi i

Strategies for constraining the size of the residual:


(6)

(7)

(8)


Slide 11

Slide 11


Galerkin method and l2-norm minimisation are equivalent when

applied to problems of the form given by equation (1). (Durran,1999)

Different series expansion methods use one or more of this strategies to minimise the error

- Collocation strategy is used in the pseudo-spectral method

- Galerkin and l2-norm minimisation are the basis of the spectral method

- Galerkin approximation is used in the finite-element method


Slide 12

Slide 12

Transform equation (1) into series-expansion form:

NjallforfRfHft jjj ,...,10)ˆ(,)ˆ(,ˆ,

=0 (Galerkin approximation)

=> NjallfordxaHdt

dadx iS

N

iij

N

i

i

S ij ,...,10)(1

*

1

*

(9)


)ˆ()ˆ(ˆ

fRfHt

f

(5)Start from equation (5) (equivalent to (1))

Take the scalar product of this equation with all the expansion functions and apply the Galerkin approximation:


Slide 13

Slide 13

Initial state in series-expansion form:

)(),( 00 xftxf Initial condition (2)

)()()(),( 01

00 xfrxtatxfN

iii

),(ˆ0txf

),(ˆ0txf Nitai ,...,1)( 0

The same strategies used for constraining the residual R in (5) can be used tominimise the error r in (10). Using the Galerkin method gives

gives the bestCompute the coefficients such that ,,...,1),( 0 Nitai ),(ˆ0txf

)(,,),(ˆ,),(, 000 xfrtxftxf jjjj

=0 (Galerkin approximation)

).(0 xfapproximation to

for all j=1,…, N.


(10)


Slide 14

Slide 14

dxxfxdxxxtaS ji

N

iS ji

)()()()()( 0*

1

*0

00 )( ftaW

01 0 ,)(, fta j

N

i iij

ijw , jf0

Initial state (cont):

for all j=1,…, N.

for all j=1,…, N.

=>


(11)

(11a)


Slide 15

Slide 15

Summary of the initial value problem (1) in series-expansion form

NjallfordxaHdt

dadx iS

N

iij

N

i

i

S ij ,...,10)(1

*

1

*

(9)

Differential equation (1):

Initial condition:

NjallfordxxfxtadxS j

N

iiS ij ,...,1)()()( 0

*

10

*

(11)

Boundary conditions:

The expansion functions have to satisfy the required conditions on the boundary of the domain S => approximated solution also satisfies these boundary conditions.



Slide 16

Slide 16

What have we gained so far from applying the series-expansion method?

Transformed the partial differential equation into ordinary differential

equations in time for the expansion coefficients ai(t).

Equations look more complicated than in finite-difference discretisation.

Main benefit from the series-expansion method comes from choosing the expansion functions judiciously depending on the form and properties of the operator H in equation (1) and on the boundary conditions.

For example, if we know the eigenfunctions ei of H, (i.e. )and we choose these as expansion functions, the problem simplifies to

iii eeH )(

Njallforadt

dadxee

N

iii

i

S ij ,...,101

*

(12)



Slide 17

Slide 17

If the functions ei are orthogonal and normalised functions, i.e.

ijiS jij dxxexeee ,* )()(,

ji ,jifor 1jifor 0

where

then the set of coupled equations (12) decouples into N independent ordinary

differential equations in t for the expansion coefficients a

(13)

(14)

Njallfortatadt

djjj ,...,10)()( (15)


Can be solved analytically!


Slide 18

Slide 18

The Spectral MethodThe Spectral Method

The feature that distinguishes the spectral method from other series-expansion methods is that the expansion functions form an orthogonal set on the domain S(not necessarily normalised).

=> the system of equations (9) reduces to

NjallfordxaHwdt

daiS

N

iij

j

j ,...,1)(1

1

*

(17)

This is a very helpful simplification, because we obtain explicit equations in when the time derivatives are replaced with finite-differences.)( tta j

jiiS ji wdxxx ,* )()( (16)

In the more general equation (9) matrix W (defined in (11)) introduces a coupling

between all the a at the new time level => implicit equations, more costly to solve!


Slide 19

Slide 19

The Spectral Method (cont)The Spectral Method (cont)

Initial condition (11) simplifies to

Njallfordxxfxw

taS j

jj ,...,1)()(

1)( 0

*0 (18)

00 )( fta

(18a)<=>

ji ,

jiforwi

1

jifor 0

Where


Slide 20

Slide 20


NjallfordxaHwdt

daiS

N

iij

j

j ,...,1)(1

1

*

(17)

Summary of the original problem (1) in spectral representation

Differential equation:

Njallfordxxfxw

taS j

jj ,...,1)()(

1)( 0

*0 (18)

Initial condition:

Boundary conditions:

The expansion functions have to satisfy the required conditions on the boundary of the domain S => approximated solution also satisfies these boundary conditions.

Expansion functions: Form an orthogonal set on the domain S

jiiS ji wdxxx ,* )()( (16)


Slide 21

Slide 21


The choice of families of orthogonal expansion functions is largely dictated by the geometry of the problem and the boundary conditions.

Examples:

Rectangular domain with periodic boundary conditions: Fourier series Non-periodic domains: (maybe) Chebyshev polynomials

Spherical geometry: spherical harmonics


Slide 22

Slide 22


In this case the spatial operator H in (1) is

Example 1: One-dimensional linear advection

Since u is constant, it is easy to find expansion functions that are eigenfunctions to H. => This problem is almost trivial to solve with the spectral method, but isa good case to reveal some of the fundamental strengths and weaknesses of thespectral method.

xuH

)()0,( 0 xfxf (initial condition) (19b)

),2(),( tnxftxf (periodic boundary condition) (19c)

0

x

fu

t

f(19a)(with constant advection velocity u)

20 x (domain S)


Slide 23

Slide 23


Example 1: One-dimensional linear advection (cont)

Fourier functions are eigenfunctions of the derivative operatorimximx eime

x

M

Mm

imxm etatxf )(),(ˆ

MmMallfortaumidt

tdam

m 0)()(

2

0

,2, mnimxinximxinx dxeeeeThey are orthogonal:

Approximate f as finite series expansion:

(20)

(21)

(22)

Insert (22) into (19a) and apply the Galerkin method => system of decoupled eqs

(23)


Slide 24

Slide 24


Example 1: One-dimensional linear advection (cont)

MmMallfortaumidt

tdam

m 0)()(

(23)

No need to discretise in time, can be solved analytically:tuim

mm eata )0()(

In physical space the solution is:

This is identical to the analytic solution of the 1d-advection problem with constant u. => Discretisation in space with the spectral method does not introduce any distortion of the solution: no error in phase speed or amplitude of the advected waves, i.e. no numerical dispersion or amplification/damping introduced, not even for the shortest resolved waves! (The centred finite-difference scheme in space introduces dispersion.)

)()0()0(),( 0)( tuxfeaeeatxf

M

Mm

tuxmim

M

Mm

xmitumim

(24)


Slide 25

Slide 25


Stability:

Integration in time by finite-difference (FD) methods when the spectral method isused for the spatial dimensions typically require shorter time steps than with FDmethods in space.This is a direct consequence of the spectral method’s ability to correctly representeven the shortest waves.

Leapfrog time integration of the 1d linear advection equation is stable if

Cx

tu

where

1C

1C

for spectral

for centred 2nd-order FD

Higher order centred FD have more restrictive CFLs,

and for infinite order FD1 C (as for spectral!)


Slide 26

Slide 26


Stability (cont):The reason behind the less restrictive CFL criterion for the centred 2nd-order FD scheme lies the misrepresentation of the shortest waves by the FD scheme:

xk

xku

)sin(

for centred 2nd-order FD

u for spectral (correct value!)

The stability criterion of the leapfrog scheme for the 1d linear advection is

1 xkck

The numerical phase speed ck is

1xu)max( kck = 1 xuuk

(FD)=>

(SP)

x

tu1

1 (FD)

(SP)


Slide 27

Slide 27


Example 2: One-dimensional nonlinear advection

)()0,( 0 xfxf (initial condition) (25b)

),2(),( tnxftxf (periodic boundary condition) (25c)

0),(

x

ftxu

t

f(25a)

20 x (domain S)

M

Mm

xkmim

k dxetxuaimdt

da

2

0

)(),(2

1

Approximate f with a finite Fourier series as in (22), insert into (25a) and apply the Galerkin procedure (alternatively, use directly (17)) =>

(26)


Slide 28

Slide 28


Example 2: One-dimensional nonlinear advection (cont)

Suppose that u is predicted simultaneously with f and given by the Fourier series

M

Mn

xinn etutxu )(),(

Inserting into (30) gives

Mnmkmn

mnk tatumi

dt

da

,,

)()( Compact, but costly to evaluate, convolution sum!

M

Mm

M

Mn

xkmnimn

k dxetatuimdt

da

2

0

)()()(2

1(27)

kmn ,2 <=>

Operation count for advancing the solution by one time step is O( (2M+1)2 ).With a finite-difference scheme only O(2M+1) arithmetic operations are required.

(27a)


Slide 29

Slide 29

The Transform Method (TM)The Transform Method (TM)

Products of functions are more easily computed in physical space while derivatives are more accurately and cheaply computed in spectral space.

It would be nice if we could combine these advantages and get the best of both worlds!

For this we need an efficient and accurate way of switching between the two spaces.

Such transformations exist: direct/inverse Fast Fourier Transforms (FFT)


Slide 30

Slide 30

The continuous Fourier Transform (FT)

Direct Fourier transform

2

0

),(2

1)( dxetxfta xim

m

Inverse Fourier transform

1

)(),(m

ximm etatxf

(28a)

(28b)

The Transform Method (cont)The Transform Method (cont)

For practical applications we need discretised analogues to these continuous Fourier transforms.


Slide 31

Slide 31

The discrete Fourier Transform

The discrete Fourier transform establishes a simple relationship between the

2M+1 expansion coefficients am(t) and 2M+1 independent grid-point values f(xj,,t) at the equidistant points

12,...,1,12

2

Mj

Mjx j

The set of expansion coefficients M

Mmm ta )(

12

1),(

M

jj txfdefines the set of grid-point values

through

M

Mm

ximmj

jetatxf )(),((discrete inverse FT)

(30b)

(29)



Slide 32

Slide 32

The discrete Fourier Transform (cont)

the set of expansion coefficients can be computed from M

Mmm ta )(

12

1),(

M

jj txfFrom the set of grid-point values And vice versa:

12

1

),(12

1)(

M

j

ximjm

jetxfM

ta(30a)

(discrete direct FT)

The relations (30a) and (30b) are discretised analogues to the standard FT (28a)and its inverse (28b). (The integral in (28a) is replaced by a finite sum in (30a).)They are known as discrete or finite (direct/inverse) Fourier transform.

mn

M

j

ximxin Mee jj

,

12

1

)12(

Valid because a discrete analogue of the orthogonality of Fourier functions (21) holds:

(31)



Slide 33

Slide 33

For the discrete Fourier transformations are exact, i.e. you can transform to spectral space and back to grid-point space and exactly recover your original function on the grid. (N: number of grid points.)

The discrete Fourier Transform (cont)

12 MN

At least 2M+1 equidistant grid-points covering the interval are needed if one want to retain in the expansion of f all Fourier modes up to the maximum wave number M.

2,0


Remarks:


Slide 34

Slide 34


Mathematically equivalent but computationally more efficient finite FTs,so-called Fast Fourier Transforms (FFTs), exist which use only O(N logN) arithmetic operations to convert a set of N grid-point values into a set of N Fourier coefficients and vice versa.

The basic idea of the transform method is to compute derivatives in spectral space (accurate and cheap!) and products of functions in grid-point space (cheap!) and use the FFTs to swap between these spaces as necessary.

The efficiency of these FFTs is key to the transform method!


Slide 35

Slide 35

Instead of evaluating the convolution sum in (27a), use the Transform Methodand proceed as follows to evaluate the product term in the non-linear advection equation (25a):

x

txftxu

t

txf

),(),(

),((25a)

We have both u and f in spectral representation:

Step 1 Compute the x-derivative of f by multiplying each am by im

M

Mmm ta )( M

Mnn tu )(

M

Mmm taimx

f

)(


Back to Example 2: One-dimensional nonlinear advection


Slide 36

Slide 36

Step 2 Use inverse FFT to transform both u and to physical space, i.e. compute the values of these two functions at the grid-points

x

f

Llxl

2

L

ll txu 1),(

L

ll txf 1' ),(

M

Mnn tu )(

M

Mmm taim )(

inv. FFT

inv. FFT

Step 3 Compute the product uf’: L

lll txftxu 1' ),(),(

Step 4 Use direct FFT to transform the product back to spectral space

L

lll txftxu 1' ),(),( direct FFT M

Mnm tfu

)('



Slide 37

Slide 37

Question: Is the result for the product obtained with the transform method identical to the product computed by evaluating the convolution sum in (27a)?

Answer: The two ways of computing the product give identical results provided aliasing errors are avoided during the computation of the product in physical space. Aliasing errors can be avoided with a sufficiently fine spatial resolution.

How fine does it have to be? How many grid points L do we need in physical space to prevent aliasing errors in the product if we have a spectral resolution with maximum wave number M?



Slide 38

Slide 38

Avoiding aliasing errors:

cl

aliasing

2MM0 wave number

21 mm m~

M is the cutoff wave number of the original expansion

xlc

2

2L

x2

is the cutoff wave number of the physical grid, where

Multiplication of two waves with wave numbers m1 and m2 gives a new wave m1+m2. If m1+m2 > lc , then m1+m2 is aliased into such that appears as the symmetricreflection of m1+m2 about the cut-off wave number lc.

m~ m~

Question: How large does lc have to be so that no waves are aliased into waves < M?

MLM

Mlc 32



Slide 39

Slide 39

A minimum of 2M+1 grid points are needed in physical space to preventaliasing errors from quadratic terms in the equations if M is the maximum retained wave number in the spectral expansion.

Such a grid is referred to as a “quadratic grid”.

Quadratic grid = grid with sufficient resolution to avoid aliasing errors from quadratic terms. Lq = 3M+1

Linear grid = grid with L = 2M+1, which ensures exact transformation of the linear terms to spectral space and back (but with aliasing errors from quadratic and higher order terms).



Slide 40

Slide 40

Viewed differently:

Given a resolution in grid-point space with L grid points, the correspondingspectral resolution (maximum wave number) is

for the linear grid

for the quadratic grid

2

1

LM l

3

1

LM q

=> 1/3 less resolution in spectral space with the quadratic grid than with thelinear grid.

Using the quadratic grid is equivalent to filtering out the 1/3 largest wave numbers, i.e. the ones that would be contaminated by aliasing errors from thequadratic terms in the equations.



Slide 41

Slide 41

The Spectral Method on the SphereThe Spectral Method on the Sphere

Spherical Harmonics Expansion

Spherical geometry: Use spherical coordinates: longitude

latitude

, sin

),(),( ,, nmm mn

nm Yaf

Any horizontally varying 2d scalar field can be efficiently represented in spherical geometry by a series of spherical harmonic functions Ym,n:

imnmnm ePY )(),( ,,

associated Legendre functions Fourier functions

(40)

(41)


Slide 42

Slide 42


Definition of the Spherical Harmonics

imnmnm ePY )(),( ,,Spherical harmonics

0),()1()!(

)!(

2

)12()( 2/2

2/1

,

mPd

d

mn

mnnP nm

mm

nm

The associated Legendre functions Pm,n are generated from the Legendre Polynomials via the expression

nn

n

nn d

d

nP )1(

!2

1)( 2

Where Pn is the “normal” Legendre

polynomial of order n defined by

This definition is only valid for ! nm

(41)

(42)

(43)

nmm

nm PP ,, )1(


Slide 43

Slide 43


Some Spherical Harmonics for n=5


Slide 44

Slide 44


Properties of the Spherical Harmonics

lnlmnm dPP ,,

1

1 , )()(

snrmsrnm ddYY ,,

1

1

*,, ),(),(

2

1

*,,,, )1()()1()( nm

mnmnm

mnm YYPP

*,, )1( nm

mnm aa

=> the expansion coefficients of a real-valued function satisfy

Spherical harmonics are orthogonal (=> suitable as basis for spectral method!):

because the Fourier functions and the Legendre polynomials form orthogonal sets:

2

0

,2 mnimxinx dxee

(44)

(45b)(45a)

(46)

(47)


Slide 45

Slide 45


Properties of the Spherical Harmonics (cont)

Eigenfunctions of the derivative operator in longitude.nmnm imY

Y,

,

,)1()1( 1,,1,1,,2

nmnmnmnmnm YnYn

Y

=> Not eigenfunctions of the derivative operator in latitude!But derivatives are easy to compute via this recurrence relation.

2/1

2

22

, 14

n

mnnm

nmnm YR

nnY ,2,

2 )1(

Eigenfunctions of the horizontal Laplace operator!Very important property of the spherical harmonics! semi-implicit time integration method leads to adecoupled set of equations for the field values at the future time-level => very easy to solve in spectralspace!

R: radius of the sphere, : horizontal Laplace operatoron the sphere. (see next page)

2

(48)

(49)

(50)


Slide 46

Slide 46


Properties of the Spherical Harmonics (cont)

Horizontal Laplacian operator on the sphere in spherical coordinates:

222

2

22

2

2

222

11

)1(

1

coscoscos

1

RR

R

(51)


Slide 47

Slide 47


Computation of the expansion coefficients

),(),(),,,( ,, nmm mn

nm Ytzatzf

2

0

),,,(2

1),,( detzftza im

m

1

1

2

0

,, )(),,,(4

1),(

ddePtzftza im

nmnm

mn

nmnmm Ptzatza )(),(),,( ,,

Fourier transform followed by Legendre transform

Performing only the Fourier transform gives the Fourier coefficients:

direct Fourier transform These Fourier coefficients can also be obtained by an inverse Legendre transform

of the spectral coefficients am,n:

inverse Legendre transform

(52)

(53)

(54)

(55)


Slide 48

Slide 48


Truncating the spherical harmonics expansion

For practical applications the infinite series in (52) must be truncated => numerical approximation of the form

),(),(ˆ,

)(

, nm

M

Mm

mN

mnnm Yaf

Examples of truncations:

m

n

M triangular rhomboidalm

n

M

MmmN )(MmN )(

Triangular truncation is isotropic, i.e. gives uniform spatial resolution over the entire surface of the sphere. Best choice of truncation for high resolution models.

(56)


Slide 49

Slide 49


The Gaussian GridWe want to use the transform method to evaluate product terms in the equations in physical space.

Need to define the grid in physical space which corresponds to the spherical harmonics expansion and ensures exact numerical transformation between spherical harmonics space and physical space.

This grid is called the Gaussian Grid.

Spherical harmonics transformation is a Fourier transformation in longitudefollowed by a Legendre transformation in latitude. =>

In longitude we use the Fourier grid with L equidistant points along a latitude circle. L=2M+1 for the linear grid L=3M+1 for the quadratic grid


Slide 50

Slide 50

Use Gaussian quardature: numerical integration method, invented by C.F. Gauss,that computes the integral of all polynomials of degree 2L-1 or less exactly from just L values of the integrand at special points called Gaussian quadrature points.

These quadrature points are the zeros of the Legendre polynomial of degree n


The Gaussian Grid (cont)

LlP GL l

,...,1,0)(,0

How large do we have to chose L? 2L-1 >= largest possible degree of the integrand in (57) (is a polynomial!)

1

1

,, )(),,(2

1),( dPtzatza nmmnm

In latitude: Legendre transform, i.e. need to compute the following integral numerically with high accuracy

(57)

(58)


Slide 51

Slide 51


The Gaussian Grid (cont)

Largest possible degree of the integrand in (57) is 2M for triangular truncation. The exact evaluation of (57) by Gaussian quadrature requires a minimum of (2M+1)/2 Gaussian quadrature points for triangular truncation.

If we want to avoid aliasing errors from quadratic terms, a minimum of (3M+1)/2 Gaussian quadrature points are required (quadratic grid).

The Gaussian quadrature points (also called Gaussian latitudes) are not equidistantly spaced, but very nearly equidistant.

Summary: The total number of grid points in the Gaussian gird on the sphere is (2M+1)(2M+1)/2 for the linear grid (3M+1)(3M+1)/2 for the quadratic grid.


Slide 52

Slide 52

Full Gaussian grid Reduced Gaussian grid

• Associated Legendre functions are very small near the poles for large m

About 30% reduction in number of points


Slide 53

Slide 53

50°N 50°N

0°

0°

Orography at T1279

10

50

100

150

200

250

300

350

400

450

500

550

600

650

684.1

50°N 50°N

0°

0°

Orography at T799

10

50

100

150

200

250

300

350

400

450

500

550

600

634.0

25 km grid-spacing( 843,490 grid-points)Current operational resolution

16 km grid-spacing(2,140,704 grid-points)Future operational resolution (from end 2009)

T799 T1279


Slide 54

Slide 54

Schematic representation of the spectral transform method in the ECMWF model

Grid-point space -semi-Lagrangian advection -physical parametrizations -products of terms

Fourier space

Spectral space -horizontal gradients -semi-implicit calculations -horizontal diffusion

FFT

LT

Inverse FFT

Inverse LT

Fourier space

FFT: Fast Fourier Transform, LT: Legendre Transform


Slide 55

Slide 55


Remarks about the Legendre transform:

1.) Expensive! Operation count: O(N2) operations at N different latitudes => O(N3)!

2.) So far there is no fast Legendre transform method available. But progress has been made recently in the mathematics community and there is hope that there might be a fast Legendre transform available soon.

3.) Lack of a fast Legendre transform makes the spherical-harmonics spectral model less efficient than spectral models that use 2d Fourier transforms.


Slide 56

Slide 56

Cost of Legendre transforms

T511T799

T1279

T2047


Slide 57

Slide 57

Comparison of cost profiles of the ECMWF model at several horizontal resolutions

0

5

10

15

20

25

30

35

40

45

50

no

rmal

ized

% c

ost

T511

T799

T1279

T2047


Slide 58

Slide 58

Advantages of the spectral representation:

a.) Horizontal derivatives are computed analytically => pressure-gradient terms are very accurate => no need to stagger variables on the grid b.) Spherical harmonics are eigenfunctions of the the Laplace operator => Solving the Helmholtz equation (arising from the semi-implicit method) is straightforward and cheap. => Applying high-order diffusion is very easy.

Disadvantage: Computational cost of the Legendre transforms is high and grows faster with increasing horizontal resolution than the cost of the rest of the model.



Slide 59

Slide 59

The Finite-Element MethodThe Finite-Element Method

The finite-element method is not widely used in atmospheric model (hyperbolicpartial differential equations), because it generates implicit equations in the unknown variables at each new time level. Costly to solve!

It is more popular in ocean modelling, because it is easily adapted toirregularly shaped domains.

It is very widely used to solve time-independent problems (e.g. in engineering)

The ECMWF model uses finite-elements for the vertical discretisation.


Slide 60

Slide 60


The expansion functions are not usually orthogonal. However, they are non-zero only on a very small section of the domain, soeach function will have overlap only with its near neighbours the resulting “overlap matrix” W (which couples the equations) will be a banded matrix => easier to handle system of coupled equations than for global basis functions.

The finite-element method is a series-expansion method which usesexpansion function with compact support (i.e. these functions are non-zeroonly on a small localised part of the domain => well suited for non-periodic or irregular domains).


Slide 61

Slide 61


1

xxj xj+1 xj+2xj-1

ej

ej+1

xj: jth node

otherwise

xxxifxx

xx

xxxifxx

xx

jjjj

j

jjjj

j

0

11

1

11

1

ej=

j

jj xex )()(

nodal values

Pieces-linear finite-elements:

Basis functions are piecewise linear functions (“hat functions”).


Slide 62

Slide 62

Cubic finite-elements: Cubic B-spline basis functions (for a regular spacing of nodes)

0

)(

)(3)(3)(3

)(3)(3)(3

)(

4

1)(

32

31

211

23

31

211

23

32

3

i

iii

iii

i

i

hhh

hhh

hB

otherwise

for

for

for

for

ii

ii

ii

ii

21

1

1

12



Slide 63

Slide 63

Cubic B-splines as basis elements

B-splines modifiedto satisfy the boundarycondition f(top)=0


Slide 64

Slide 64

0

( ) ( )F f x dx

can be approximated as

Applying the Galerkin method with test functions tj =>

-1 AC Bc C A Bc

2 2

1 1 0

( ) ( )

K M

i i i ii K i M

C d c e x dx

Basis sets

2 2

1 1

1 1

1 2

0 0 0

( ) ( ) ( ) ( ) for

xK M

i j i i j ii K i M

C t x d x dx c t x e y dy dx N j N

Aji Bji

The Finite-Element Scheme in the ECMWF modelThe Finite-Element Scheme in the ECMWF model


Slide 65

Slide 65

1 -c S f

F PC

1 -1 -F P A B S f J f

Including the transformation from grid-point (GP) representation tofinite-element representation (FE)

and the projection of the result from FE to GP representation

one obtains

Matrix J depends only on the choice of the basis functions and the level spacing. It does not change during the integration of the model, so it needs to be computedonly once during the initialisation phase of the model and stored.



Slide 66

Slide 66

Cubic B-splines as basis elements

Basis elementsfor the represen-tation of thefunction tobe integrated(integrand)

f

Basis elementsfor the representationof the integral

F


Slide 67

Slide 67

High order accuracy (8th order for cubic elements)

Very accurate computation of the pressure-gradient term

in conjunction with the spectral computation of

horizontal derivatives

More accurate vertical velocity for the semi-Lagrangian

trajectory computation

Reduced vertical noise in the model

No staggering of variables required in the vertical: good for semi-Lagrangian scheme because winds and advected variables are represented on the same vertical levels.

Benefits from using this finite-element scheme in the vertical in the ECMWFmodel:



Slide 68

Slide 68

Thank you very much for your attention.

Documents

Slide 1 Numerical Methods: Series-Expansion Methods Slide 1 Agathe Untch e-mail: [email protected] (office 011) Series-Expansion Methods