Offenbach 11.3.2003

Implicit Time Integration andNH Schemes of the Next

GenerationJ. Steppeler, DWD

Offenbach 11.3.2003

• Accuracy/high (Spatial) Approximation order

• Z-Coordinate

• Efficiency: High Order of Time Integration; Two Time Levels; Serendipidity Grids; “Large” Time Steps; Implicit Time Integration

Some Desirable features of Next Generation NH Models

Offenbach 11.3.2003

• Z-Coordinate/3rd Order Spatial Approximation/Serendipidity Grids

• Different NH Approaches

• Solution of non-Periodic Problems by Generalised Fourier Transform

• Efficiency Considerations

• Computational Example of a non-Periodic Problem

Plan of Lecture

Offenbach 11.3.2003

Offenbach 11.3.2003

Ilder Heinz-Werner

3-D Cloud-Picture

18 January1998

The mountain related bias of convectional clouds and precipitation is supposed to disappear with the Z-coordinate

Offenbach 11.3.2003

i - 1/2

i - 1/2

j - 1/2

j + 1/2

j - 1/2

i + 1/2

i + 1/2j + 1/2

i, j

Shaved Elements

Offenbach 11.3.2003

Cross Section for Flow Over the Alps, Forecasted by Eta Model with Step Orography (Right) and with Terrain Following Orography (Left)

Offenbach 11.3.2003

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The geometry of 1st, 2nd and 3rd order representation of the (x) field

Offenbach 11.3.2003

Third Order finite element representation(Continuous cubic spline)

Basis functions:

1 - d:

Auxiliary grid point

Main grid point

Field represention: Field operators:

• All classical grid operations• Operations using the polynomial derivatives

(inhomogeneous grid operations)• Galerkin operations

v

vivi b ,,

Offenbach 11.3.2003

Order consistent field representations in one and three space dimensions

Order consistent interpolation requires the field only in the solid lines

Offenbach 11.3.2003

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The continuous splines in the presence of Orography:

(x) must be a „ smooth“ function (not a step function)

= points to pose boundary conditions.The dashed parts of the grid lines are

used to pose boundary values.

Offenbach 11.3.2003

Implicit Approaches

2/)( : dhomogeniseLocally

2/)( :linear Tangent

2/)( : Nonlinear

field dynamic :h field;constant :u )( :Example

11

111

11

nx

nx

nn

xn

xnn

xn

xnn

nx

nx

nn

uhuhdthh

huhuhuhudthh

uhuhdthh

xuh

th

Offenbach 11.3.2003

• The LH - is the Most Common SI Method

• The Equations of Motion are homogeneously linearised at each Grid Point

• At Each Grid-Point a Problem of Constant Coefficients is Defined

• For Each Grid-Point The Associated Linear Problem Can be Solved Using an FT and a Linear Problem Specific to Each Grid-Point

• The GFT (Generalised FT) Computes the Results of the Different FTs Using One Transform

• The numerical cost of GFT is Simlar to that of an FT

• A Fast GFT exists similar to Fast FT

Direct Methods for Locally Homogenized SI

Offenbach 11.3.2003

Split Representation of Derivatives xxx

,0

0

0)()( 10

0

xx

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• The Fourier Coefficients are the same for the grid Points of a Subregion

• The Linearised Eqs. are different for each Gridpoint

• In Case of only One Subregion the Support!Points of the derivative are the Boundary Values

• does not create time-Step Limitations

• GFT Returns the Grid-Point-Values after doing a Different Eigenvalue Calculation at Each Point

Organisation of the Implicit Time-Step

x0

x0

Offenbach 11.3.2003

The SI Timestep Subtract Large Scale Part of each function

Compute Fourier

Coefficients

Choose Gridpoint

Compute next time level in

Fourier Space

Transform Back

Use Result Only for Chosen Grid-Point

Do the Above for all Other Grid Points

Set Boundary Values as in Finite Difference Methods

Offenbach 11.3.2003

Example:1-d Schallow Water Eq

)2/)(2/)((

)2/)(2/)((

:Scheme SI

)(

111

111

nx

nx

nnx

nx

nnn

nx

nx

nx

nx

nnn

hhuuuhdthh

hhuuudtuu

xhu

th

x

h

x

uut

u

Definitions:nx

nnx

nnx

nx

n

nnnn

uhhurshhuursu

hhhuuu

;

;; 11

SI Scheme:x

nx

nx

nx

nx

nx

n

xxn

xxn

xxn

udthhdtuuhhudtrshudthhdturshh

hdtudtuhuudtrsuhdtudtursuu

'')(

'')(00

00

Periodicity Terms (=0in the following)

Offenbach 11.3.2003

Boundary- and Exterior Points

Point to pose (Artificial) Boundary Values

Redundant Points

•Redundant points can be included in the FT

•The result of the time-step does not depend on the continuation of the field to redundant points

Offenbach 11.3.2003

Operation in Fourier Space

Definition:

SI Scheme:gridpoint.chosen someat taken be will,,, following In the

''

''

nn

xn

xn

xn

xn

xxn

xxn

hurshrsu

udthhdturshudthhdturshh

hdtudtursuhdtudtursuu

F~

Linear Equations at the chosen gridpoint:

only. gridpoint chosen at the taken be will,, fields formed back trans The

equation.linear theof solution thefrom obtained are ,

~~1~~

~~

~~~~

~~~~

1

hFuFhu

hu

rshudthhhdtu

rsuhdtudtuu

xn

xn

xxn

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Computational Example: 1-d Schock Wave

The time Step could be increased up to the CFL of advection (10 m/sec)

Offenbach 11.3.2003

• A direct si- method was proposed• The method is based on a generalised Fourier

Transform• The generalised FT is potentially as efficient as

the FT (fast FT)• The method is efficient for increased spatial order• 1-d tests have been performed

Conclusions