Offenbach 11.3.2003 Semi-Implicit Time Integration (NH Schemes of the Next Generation) J. Steppeler,...
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Transcript of Offenbach 11.3.2003 Semi-Implicit Time Integration (NH Schemes of the Next Generation) J. Steppeler,...
Offenbach 11.3.2003
Semi-Implicit Time Integration (NH Schemes of the Next
Generation)J. 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
Offenbach 11.3.2003
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
Offenbach 11.3.2003
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/))()((: d homogeniseLocally
2/))()((
2/))()(( :linear Tangent
2/)( : Nonlinear
field dynamic :h field;constant :u )( :Example
111
11
111
11
nnx
nx
nx
nx
nnn
xnnnnn
xn
nnx
nx
nx
nx
nnn
nx
nx
nn
huuhhudthh
huuhhudt
huuhhudthh
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
Offenbach 11.3.2003
• 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~~
~~
~~~~
~~~~
hFuFhu
hu
rshudthhhdtu
rsuhdtudtuu
xn
xn
xxn
Offenbach 11.3.2003
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