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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