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RITM – codeComputation with stiff transport models
presented by D.Kalupin
12th Meeting of the ITPA Transport Physics (TP) Topical Group 7-10 May 2007, EPFL, Lausanne, Switzerland
Institut für Plasmaphysik, Forschungszentrum Jülich GmbH, EURATOM Association, D-52425 Jülich, Germany
EFDA Integrated Tokamak Modelling (ITM) Task Force
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-There is a number of stiff transport models (GLF23, RITM, Weiland) used in European codes (ASTRA, CRONOS, JETTO and RITM)
- New transport code, which is under development by the EFDA ITM Task Force, and will be to a large extend assembled from existing codes, should be capable of working with stiff models
- Thus, methods of reliable, stable operation with stiff models is one of current and urgent tasks for the ITM-TF
- RITM code has a long time experience of operation with transport coefficients being strongly non-linear functions of of plasma parameter gradients
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Why stiff transport models cause problems? General form of transport equation:
,,12
1
FvFDSggt
F
After time discretization:
SFvFDgg
FF t
21
1 1
With stiff transport models D and are strongly non-linear functions of gradients of
plasma parameters, terms and can lead to numerical instabilities
Dv
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Methods to avoid these numerical instabilities used in present codes
- reduction of the time step usually, the time step is reduced down to 10-5-10-4 s, or even 10-6 s, which requires large simulation time
- smoothing of profiles and/or transport coefficients a plenty of smoothing routines is developed and used in different codes, (caution: such procedure can smooth away important physics)
- reformulation of transport equations integral form of transport equations does not contain the radial derivatives of transport coefficients
- developments to solvers
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RITM approach to solve transport equations
Standard form of transport equation: SFvFDgg
FF t
21
1 1
dgFSFvFDgdgF t
10
1
20
1
NdFg
0
12
1 JdgFSt
10
1
2
1
DgJgNg
gDv
Dggg
gDvNN
2
11
12
11
12
2 1213
1
2
g
NNF
RITM integral form:
New variables:
New differential equation for N does not include derivatives of transport coefficients and any assumption about the function behavior in a grid sell, that improves the convergence
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Numerical approach for solving of differential equations in RITM
Ordinary differential second order equations for all variables: Between any two neighboring grid knots iiii hrrr 1,
and y0 is linearly interpolated
2,
211 iiii bbbaaa
ryyrbdrdyra
dryd
02
2
Introducing new variable
iii
iii rr
rryyyry
10
10
00
ii
ii
rryy
bayyy
10
10
01
11
21
2
yrbdrdyra
dryd
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Number of grid points, N
Tota
l num
ber o
f ite
ratio
nsne
eded
to a
chie
ve a
ste
ady
stat
e
Numerical approach for solving of differential equations in RITM
Tokar et.al, Computer Phys. Communications,175 (2006) 30-35
rFCrFCrrry iiiiii
i221111
Coefficients are determined
from continuity of in knots
ii CC 21 ,
drdyy,
and from boundary conditions
This homogeneous equation hasKnown analytical solutions:
This approach requires less iterations to get the steady state solution and allows to obtain solution with enough accuracy for larger time step
RITM solver
finite volume
Time step: M
t statesteady _
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Smoothing routines …Diverse smoothing routines have been impemented in RITM
2. Smoothing by parabolic curve:
1. Fitting by smooth spline function which consists of three segments:
!!! Should be used carefully, as it can smooth away the physics…
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RITM transport model CORE TRANSPORT EDGE TRANSPORT
eeff
effD
Teeff
tr Ten
ii
ifn
~
1~*
0~~,
drdnVni ri
i
iD
eTi T
Tnn
Te
~
35~
32~
32
*
cBj
nTTikcBj
Vnmi rie
yrii
~~,
~~
,
0~~,
drdnVni ri
0~~
~||
rjjik
lj
j ryy
ej
mBB
rnT
nTEenVmi eiere
eee||
0||||||,
~~~~~
||||||22||~4~,~~4~ j
ckiBj
ckiE
yr
y
0Re,
if
k ITGITG
0Re,
if
kTETE
edgeedge k,
!!! Resulting transport coefficients are strongly non-linear functions of gradients of plasma parameters
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0.0 0.2 0.4 0.6 0.8 1.00
1
2
3
4
normalised minor radius
n e, 1013
cm-3
0.0
0.5
1.0
1.5
2.0
Ti , keV
0.80 0.85 0.90 0.95 1.000
2
4
6
8
D
m2 s-1
normalised minor radius
Transport simulationsUsing RITM approach, numerical instabilities due to very fast change (in space and time) of transport coefficients are completely avoided
The time step in simulations can be increased up to 0.1 s and the space resolution can be up to 1000 radial points
The total CPU time consumption is several times smaller than with the standard approach RITM run (201 radial points) ~ 30 minJETTO run (101 radial points) ~ 5 hours
Predictive calculations of L-H transition in TEXTOR with RITM
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Summary
RITM approach to solve transport equations allows for using of stiff transport models and avoid numerical instabilities due to very fast change of transport coefficients
This method can be applied for calculations with sufficiently larger time step
It is going to be one of the options for the core transport solver in ITM-TF