1. Series-Expansion Methods
Agathe Untch
(office 011)
Numerical Methods: Series-Expansion Methods

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)
Numerical Methods: Series-Expansion Methods

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.
Numerical Methods: Series-Expansion Methods

4. Grid-point space Spectral space
Schematic representation of the spectral transform method in the ECMWF model
Grid-point space
-semi-Lagrangian advection
-physical parametrizations
-products of terms
FFT
Inverse FFT
Fourier space
Fourier space LT
Inverse LT
Spectral space
-horizontal gradients
-semi-implicit calculations
-horizontal diffusion
FFT: Fast Fourier Transform, LT: Legendre Transform
Numerical Methods: Series-Expansion Methods

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
Numerical Methods: Series-Expansion Methods

6. Fundamentals of series-expansion methods
We demonstrate the fundamentals of series-expansion methods on the following
problem for which we seek solutions:
Partial differential equation:
(with an operator H involving
only derivatives in space.)
(1)
To be solved on the spatial domain S subject to specified initial and boundary
conditions.
Initial condition:
(1a)
Boundary conditions: Solution f has to fulfil some specified conditions
on the boundary of the domain S.
Numerical Methods: Series-Expansion Methods

7. Fundamentals of series-expansion methods (cont)
The basic idea of all series-expansion methods is to write the spatial dependence
of f as a linear combination of known expansion functions
(3)
should span the L2 space, i.e. a Hilbert
The set of expansion function
space with the inner (or scalar) product of two functions defined as
(4)
The expansion functions should all satisfy the required boundary conditions.
Numerical Methods: Series-Expansion Methods

8. Fundamentals of series-expansion methods (cont)
Numerically we can’t handle infinite sums. Limit the sum to a finite number
of expansion terms N
(3a)
is only an approximation to the true solution f of the equation (1).
=>
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.
Numerical Methods: Series-Expansion Methods

9. Fundamentals of series-expansion methods (cont)
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:
(5)
(1)
<=>
R = amount by which the approximate solution fails to satisfy the
governing equation.
Strategies for minimising the residual R:
1.) minimise the l2-norm of R
2.) collocation strategy
3.) Galerkin approximation
(Details on the next slide.)
Each of these strategies leads to a system of N coupled ordinary differential
equations for the time-dependent coefficients ai(t), i=1,…, N.
Numerical Methods: Series-Expansion Methods

10. Fundamentals of series-expansion methods (cont)
Strategies for constraining the size of the residual:
1.) Minimisation of the l2-norm of the residual
Compute the ai(t) such as to minimise
(6)
2.) Collocation method
Constrain the residual by requiring it to be zero at a discrete set of grid points
(7)
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.
(8)
Numerical Methods: Series-Expansion Methods

11. Fundamentals of series-expansion methods (cont)
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
Numerical Methods: Series-Expansion Methods

12. Fundamentals of series-expansion methods (cont)
Transform equation (1) into series-expansion form:
Start from equation (5) (equivalent to (1))
(5)
Take the scalar product of this equation with all the expansion functions and
apply the Galerkin approximation:
=0 (Galerkin approximation)
(9)
=>
Numerical Methods: Series-Expansion Methods

13. Fundamentals of series-expansion methods (cont)
Initial state in series-expansion form:
Initial condition
(2)
(10)
gives the best
Compute the coefficients
such that
approximation to
The same strategies used for constraining the residual R in (5) can be used to
minimise the error r in (10). Using the Galerkin method gives
for all j=1,…, N.
=0 (Galerkin approximation)
Numerical Methods: Series-Expansion Methods

14. Fundamentals of series-expansion methods (cont)
Initial state (cont):
for all j=1,…, N.
for all j=1,…, N.
(11)
=>
(11a)
Numerical Methods: Series-Expansion Methods

15. Fundamentals of series-expansion methods (cont)
Summary of the initial value problem (1) in series-expansion form
Differential equation (1):
(9)
Initial condition:
(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.
Numerical Methods: Series-Expansion Methods

16. Fundamentals of series-expansion methods (cont)
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
(12)
Numerical Methods: Series-Expansion Methods

17. Fundamentals of series-expansion methods (cont)
If the functions ei are orthogonal and normalised functions, i.e.
(13)
where
(14)
then the set of coupled equations (12) decouples into N independent ordinary
differential equations in t for the expansion coefficients a
(15)
Can be solved analytically!
Numerical Methods: Series-Expansion Methods

18. The 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).
(16)
=> the system of equations (9) reduces to
(17)
This is a very helpful simplification, because we obtain explicit equations in
when the time derivatives are replaced with finite-differences.
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!
Numerical Methods: Series-Expansion Methods

19. The Spectral Method (cont)
Initial condition (11) simplifies to
(18)
(18a)
<=>
Where
Numerical Methods: Series-Expansion Methods

20. The Spectral Method (cont)
Summary of the original problem (1) in spectral representation
Expansion functions: Form an orthogonal set on the domain S
(16)
Differential equation:
(17)
Initial condition:
(18)
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.
Numerical Methods: Series-Expansion Methods

21. The Spectral Method (cont)
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
Numerical Methods: Series-Expansion Methods

22. The Spectral Method (cont)
Example 1: One-dimensional linear advection
(19a)
(with constant advection velocity u)
(domain S)
(initial condition)
(19b)
(periodic boundary condition)
(19c)
In this case the spatial operator H in (1) is
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 is
a good case to reveal some of the fundamental strengths and weaknesses of the
spectral method.
Numerical Methods: Series-Expansion Methods

23. The Spectral Method (cont)
Example 1: One-dimensional linear advection (cont)
Fourier functions are eigenfunctions of the derivative operator
(20)
They are orthogonal:
(21)
Approximate f as finite series expansion:
(22)
Insert (22) into (19a) and apply the Galerkin method => system of decoupled eqs
(23)
Numerical Methods: Series-Expansion Methods

24. The Spectral Method (cont)
Example 1: One-dimensional linear advection (cont)
(23)
No need to discretise in time, can be solved analytically:
In physical space the solution is:
(24)
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.)
Numerical Methods: Series-Expansion Methods

25. The Spectral Method (cont)
Stability:
Integration in time by finite-difference (FD) methods when the spectral method is
used for the spatial dimensions typically require shorter time steps than with FD
methods in space.
This is a direct consequence of the spectral method’s ability to correctly represent
even the shortest waves.
Leapfrog time integration of the 1d linear advection equation is stable if
for spectral
where
for centred 2nd-order FD
Higher order centred FD have more restrictive CFLs,
and for infinite order FD
(as for spectral!)
Numerical Methods: Series-Expansion Methods

26. The Spectral Method (cont)
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:
for centred 2nd-order FD
The numerical phase speed ck is
for spectral (correct value!)
The stability criterion of the leapfrog
scheme for the 1d linear advection is
(FD)
(FD)
=> =
(SP)
(SP)
Numerical Methods: Series-Expansion Methods

27. The Spectral Method (cont)
Example 2: One-dimensional nonlinear advection
(25a)
(domain S)
(initial condition)
(25b)
(periodic boundary condition)
(25c)
Approximate f with a finite Fourier series as in (22), insert into (25a) and apply
the Galerkin procedure (alternatively, use directly (17)) =>
(26)
Numerical Methods: Series-Expansion Methods

28. The Spectral Method (cont)
Example 2: One-dimensional nonlinear advection (cont)
Suppose that u is predicted simultaneously with f and given by the Fourier series
Inserting into (30) gives
(27)
<=>
(27a)
Compact, but costly to
evaluate, convolution sum!
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.
Numerical Methods: Series-Expansion Methods

29. 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)
Numerical Methods: Series-Expansion Methods

30. The Transform Method (cont)
The continuous Fourier Transform (FT)
Direct Fourier transform
(28a)
Inverse Fourier transform
(28b)
For practical applications we need discretised analogues to these continuous
Fourier transforms.
Numerical Methods: Series-Expansion Methods

31. The Transform Method (cont)
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
(29)
The set of expansion coefficients
defines the set of grid-point values
through
(30b)
(discrete inverse FT)
Numerical Methods: Series-Expansion Methods

32. The Transform Method (cont)
The discrete Fourier Transform (cont)
the set of expansion coefficients can be computed from
From the set of grid-point values
And vice versa:
(30a)
(discrete direct FT)
Valid because a discrete analogue of the
orthogonality of Fourier functions (21) holds:
(31)
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.
Numerical Methods: Series-Expansion Methods

33. The Transform Method (cont)
The discrete Fourier Transform (cont)
Remarks:
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.
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.)
Numerical Methods: Series-Expansion Methods

34. The Transform Method (cont)
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 efficiency of these FFTs is key to the transform method!
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.
Numerical Methods: Series-Expansion Methods

35. The Transform Method (cont)
Back to Example 2: One-dimensional nonlinear advection
Instead of evaluating the convolution sum in (27a), use the Transform Method
and proceed as follows to evaluate the product term in the non-linear
advection equation (25a):
(25a)
We have both u and f in spectral representation:
Step 1 Compute the x-derivative of f by multiplying each am by im
Numerical Methods: Series-Expansion Methods

36. The Transform Method (cont)
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
inv. FFT
Step 3 Compute the product uf’:
Step 4 Use direct FFT to transform the product back to spectral space
direct FFT
Numerical Methods: Series-Expansion Methods

37. The Transform Method (cont)
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?
Numerical Methods: Series-Expansion Methods

38. The Transform Method (cont)
Avoiding aliasing errors:
aliasing 2M M
wave number
M is the cutoff wave number of the original expansion
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 symmetric
reflection of m1+m2 about the cut-off wave number lc.
Question: How large does lc have to be so that no waves are aliased into waves < M?
Numerical Methods: Series-Expansion Methods

39. The Transform Method (cont)
A minimum of 2M+1 grid points are needed in physical space to prevent
aliasing 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).
Numerical Methods: Series-Expansion Methods

40. The Transform Method (cont)
Viewed differently:
Given a resolution in grid-point space with L grid points, the corresponding
spectral resolution (maximum wave number) is
for the linear grid
for the quadratic grid
=> 1/3 less resolution in spectral space with the quadratic grid than with the
linear 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 the
quadratic terms in the equations.
Numerical Methods: Series-Expansion Methods

41. The Spectral Method on the Sphere
Spherical Harmonics Expansion
Spherical geometry: Use spherical coordinates: longitude
latitude
Any horizontally varying 2d scalar field can be efficiently represented in
spherical geometry by a series of spherical harmonic functions Ym,n:
(40)
(41)
associated
Legendre functions
Fourier functions
Numerical Methods: Series-Expansion Methods

42. The Spectral Method on the Sphere
Definition of the Spherical Harmonics
Spherical harmonics
(41)
The associated Legendre functions Pm,n are generated from the Legendre
Polynomials via the expression
(42)
Where Pn is the “normal” Legendre
polynomial of order n defined by
(43)
This definition is only valid for !
Numerical Methods: Series-Expansion Methods

43. The Spectral Method on the Sphere
Some Spherical Harmonics for n=5
Numerical Methods: Series-Expansion Methods

44. The Spectral Method on the Sphere
Properties of the Spherical Harmonics
Spherical harmonics are orthogonal (=> suitable as basis for spectral method!):
(44)
because the Fourier functions and the Legendre polynomials form orthogonal
sets:
(45a)
(45b)
(46)
=> the expansion coefficients of a real-valued function satisfy
(47)
Numerical Methods: Series-Expansion Methods

45. The Spectral Method on the Sphere
Properties of the Spherical Harmonics (cont)
(48)
Eigenfunctions of the derivative operator in longitude.
(49)
=> Not eigenfunctions of the derivative operator in latitude!
But derivatives are easy to compute via this recurrence relation.
(50)
Eigenfunctions of the horizontal Laplace operator!
Very important property of the spherical harmonics!
semi-implicit time integration method leads to a
decoupled set of equations for the field values at the
future time-level => very easy to solve in spectral
space!
R: radius of the sphere,
: horizontal Laplace operator
on the sphere. (see next page)
Numerical Methods: Series-Expansion Methods

46. The Spectral Method on the Sphere
Properties of the Spherical Harmonics (cont)
Horizontal Laplacian operator on the sphere in spherical coordinates:
(51)
Numerical Methods: Series-Expansion Methods

47. The Spectral Method on the Sphere
Computation of the expansion coefficients
(52)
(53)
Fourier transform followed by Legendre transform
Performing only the Fourier transform gives the Fourier coefficients:
(54)
direct Fourier transform
These Fourier coefficients can also be obtained by an inverse Legendre transform
of the spectral coefficients am,n:
(55)
inverse Legendre transform
Numerical Methods: Series-Expansion Methods

48. The Spectral Method on the Sphere
Truncating the spherical harmonics expansion
For practical applications the infinite series in (52) must be truncated
=> numerical approximation of the form
(56)
Examples of truncations: m n M
triangular
rhomboidal m n M
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.
Numerical Methods: Series-Expansion Methods

49. The Spectral Method on the Sphere
The Gaussian Grid
We 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 longitude
followed 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
Numerical Methods: Series-Expansion Methods

50. The Spectral Method on the Sphere
The Gaussian Grid (cont)
In latitude: Legendre transform, i.e. need to compute the following integral
numerically with high accuracy
(57)
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
(58)
How large do we have to chose L?
2L-1 >= largest possible degree of the integrand in (57) (is a polynomial!)
Numerical Methods: Series-Expansion Methods

51. The Spectral Method on the Sphere
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.
Numerical Methods: Series-Expansion Methods

52. Full Gaussian grid Reduced Gaussian grid
About 30% reduction in number of points
• Associated Legendre functions are very small near the poles for large m
Numerical Methods: Series-Expansion Methods

53. T799 T1279 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)
Numerical Methods: Series-Expansion Methods

54. Grid-point space Spectral space
Schematic representation of the spectral transform method in the ECMWF model
Grid-point space
-semi-Lagrangian advection
-physical parametrizations
-products of terms
FFT
Inverse FFT
Fourier space
Fourier space LT
Inverse LT
Spectral space
-horizontal gradients
-semi-implicit calculations
-horizontal diffusion
FFT: Fast Fourier Transform, LT: Legendre Transform
Numerical Methods: Series-Expansion Methods

55. The Spectral Method on the Sphere
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.
Numerical Methods: Series-Expansion Methods

56. Cost of Legendre transforms
Numerical Methods: Series-Expansion Methods

57. Comparison of cost profiles of the ECMWF model at several horizontal resolutions
Numerical Methods: Series-Expansion Methods

58. The Spectral Method on the Sphere
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.
Numerical Methods: Series-Expansion Methods

59. The Finite-Element Method
The finite-element method is not widely used in atmospheric model (hyperbolic
partial 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 to
irregularly 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.
Numerical Methods: Series-Expansion Methods

60. The Finite-Element Method
The finite-element method is a series-expansion method which uses
expansion function with compact support (i.e. these functions are non-zero
only on a small localised part of the domain => well suited for non-periodic
or irregular domains).
The expansion functions are not usually orthogonal.
However, they are non-zero only on a very small section of the domain, so
each 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.
Numerical Methods: Series-Expansion Methods

61. The Finite-Element Method
Pieces-linear finite-elements:
Basis functions are piecewise linear functions (“hat functions”). 1 x xj
xj+1
xj+2
xj-1 ej
ej+1
xj: jth node
ej=
nodal values
Numerical Methods: Series-Expansion Methods

62. The Finite-Element Method
Cubic finite-elements: Cubic B-spline basis functions
(for a regular spacing of nodes)
Numerical Methods: Series-Expansion Methods

63. Cubic B-splines as basis elements B-splines modified
to satisfy the boundary
condition f(top)=0
Cubic B-splines as
basis elements
Numerical Methods: Series-Expansion Methods

64. The Finite-Element Scheme in the ECMWF model
can be approximated as
Basis sets
Applying the Galerkin method with test functions tj =>
Aji
Bji
Numerical Methods: Series-Expansion Methods

65. The Finite-Element Scheme in the ECMWF model
Including the transformation from grid-point (GP) representation to
finite-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 computed
only once during the initialisation phase of the model and stored.
Numerical Methods: Series-Expansion Methods

66. Cubic B-splines as basis elements Basis elements Basis elements
for the represen-
tation of the
function to
be integrated
(integrand) f
Basis elements
for the
representation
of the integral F
Numerical Methods: Series-Expansion Methods

67. The Finite-Element Scheme in the ECMWF model
Benefits from using this finite-element scheme in the vertical in the ECMWF
model:
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.
Numerical Methods: Series-Expansion Methods

68. Thank you very much for your attention.
Numerical Methods: Series-Expansion Methods