1 A new iterative technique for solving nonlinear coupled equations arising from nuclear waste transport processes H. HOTEIT 1,2, Ph. ACKERER 2, R. MOSE.

1

A new iterative technique for solving nonlinear coupled equations arising

from nuclear waste transport processes

H. HOTEIT 1,2, Ph. ACKERER2, R. MOSE2,3

1IRISA-INRIA, Rennes

2Institut de Mécanique des Fluides et des Solides, IMFS, Strasbourg

3Ecole Nationale du Génie de l'Eau et de l'Environnement, ENGEES, Strasbourg

34ème Congrès National d'Analyse Numérique

27 Mai - 31 Mai 2002

2

Outline

Mathematical model of the transport processes.

Numerical methods:

Mixed Hybride Finite Element method (MHFE);

Discontinuous Galerkin method (DG).

Linearization techniques:

Picard (fixed point) method;

Newton-Raphson method.

Some numerical results.

3

Transport Processes

The transport process concerns an isolated nuclide chain :

U Pu 234238

with the following transport mechanisms :

advection, dispersion/diffusion ;

mass production/reduction ;

precipitation/dissolution ;

simplified chemical reactions (sorption).

4

kkkkkkkkkkk

k SCRCRuCCDt

CR

''').(

kkksolkksolk

ksol SFFt

FR

'')1()1()1(

Mathematical model

Transport equation

Sk is a nonlinear precipitation/dissolution term

satkkk

satk

ksatkk

k

CCifCC

FandCCifS

00

)1(

)1(

2,1

)(

kksolkk

ksolkksatke

satk FCR

FCRCC

5

Numerical methods

Operator splitting technique is used by coupling

Diffusion/dispersion by MHFEM

Advection by DGM

Linearization is done by using

Picard (Fixed Point) method

Newton-Raphson method

6

MHFEAdvantages

mass is conserved locally ;

the state head and its gradient are approximated simultaneously ;

velocity is determined everywhere due to Raviart-Thomas space functions; full tensors of permeability are easily approximated ; Fourier BC are easily handled ; it can be simply extended to unstructured 2D and 3D grids ;the linear system to solve is positive definite.

Disadvantages scheme is non monotone ;number of degrees of freedom=number of sides (faces).

7

DGM

Advantages mass is conserved locally ; satisfies a maximum principle (conserves the positively of the solution) ; can capture shocks without producing spurious oscillation ;

ability to handle complicated geometries ;

simple treatment of boundary conditions.

Disadvantages

limited choice of the time-step (explicit time discretization) ;slope (flux) limiting operator stabilize the scheme

but creates small amount of numerical diffusion.

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''' ,,,,)1( kkkkkkkkk

ksol FFCCSFFAt

FR

''' ,,,, kkkkkkkkk

k FFCCSCCBt

CR

Linearization by the Picard method

The transport system is rewritten in the form

where,

''' )1()1(, kksolkksolkkk FFFFA

'''' ).(),( kkkkkkkkkkkk CRCRuCCDCCB

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Linearization by the Picard method

The (m+1)th step of the Picard-iteration process

Stopping criteria

2

1,1

2

,11,1

2

1,1

2

,11,1

mnkba

mnk

mnk

mnkba

mnk

mnk

FFF

CCC

1'

1,11'

,11'

1,11,1

1'

,11'

1,11'

,11,1

,*,,,

,,,,*

)1(

nk

mnk

nk

mnkk

nk

mnkk

nk

mnk

k

nk

mnk

nk

mnkk

nk

mnkk

nk

mnk

ksol

FFCCSCCBt

CCR

FFCCSFFAt

FFR

10

Linearization by the Picard method

6 7 8 9 10 11 12 13 14 15

0.0

4.0x10-7

8.0x10-7

1.2x10-6

1.6x10-6

2.0x10-6

2.4x10-6

2.8x10-6

Cri

tère

de

co

nve

rge

nce

No d'itération

Erreur sur Fk

Erreur sur Ck

Convergence needs very small time steps, otherwise :

Residual errors for C and F

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Coupling Picard and Newton-Raphson methods

1'

,11'

,11'

,1,1

,1 ,,,,)1(

n

kmn

knk

mnkk

nk

mnkk

nk

mnk

ksolmn

k FFCCSFFAt

FFRF

Define the residual function

)()( ,1

,1

,1mn

km

kmnk

mnk FF

F

F

By using Taylor’s approximation , we get

By simple differentiating, we obtain

)( ,1,1

,1

,1

,1

,1mn

kmn

kmnk

mnk

mnk

mnkksol FF

F

S

F

A

t

R

12

Coupling Picard and Newton-Raphson methods

The iterative process

2

21

1

1

for)1(

f

f)1(

nmt

nmnort

nmort

tn

n

n

n

10

Time steps

1'

1,11'

,11'

1,11,1

,11,1

,1

1

,1

,1

,1

,1,1

,,,,

)(

nk

mnk

nk

mnkk

nk

mnkk

nk

mnk

k

mnk

nk

mnk

mnkmn

k

mnk

mnk

mnkksolmn

k

FFCCSCCBt

CCR

FFF

FF

S

F

A

t

RF

13

3 4 5 6 7 8

0,0

1,0x10-10

2,0x10-10

3,0x10-10

4,0x10-10

5,0x10-10

Erreur sur Fk

Erreur sur Ck

Cri

tère

de

co

nve

rge

nce

No d'itérations

Coupling Picard and Newton-Raphson methods

Convergence is attained even with bigger time steps (20 times bigger)

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Some numerical results

20

40

60

80

100

Z

0

24.8

49.6

X

4.513.5Y

XY

Z

Repository site Network of alveolus Elementary cell

The repository is made up of a big number of alveolus.

Computation is made on an elementary cell .

Periodic boundary conditions are used .

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106 years105 years104 years

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0 1x102 2x102 3x102 4x102 5x102 6x102 7x102 8x102 9x102 1x103

0.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

6.0x10-4

7.0x10-4

8.0x10-4

9.0x10-4

1.0x10-3

Masse dissoute Masse precipité

234 U

( m

ol )

Temps ( années )

0.0 2.0x105 4.0x105 6.0x105 8.0x105 1.0x106

0.00

0.01

0.02

0.03

0.04

0.05

0.06 Masse dissoule + masse présipitée Masse injéctée

234 U

( m

ol )

Temps ( années )

0.0 2.0x105 4.0x105 6.0x105 8.0x105 1.0x106

0.0

3.0x10-5

6.0x10-5

9.0x10-5

1.2x10-4

1.5x10-4

Err

eur r

elat

ive

Temps ( années )

Precipitated and dissolved mass in the domain

Mass balance in the domain Relative error after 106 years

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Conclusion

Coupling DG and MHEF methods to solve a transport equation with nonlinear precipitation /dissolution function .

By using the Picard method, small time steps should be considered otherwise no convergence is attained.

Coupling Picard and Newton-Raphson methods

Newton-Raphson methods is used for solid phase equation.

Picard method methods is used for the transport equation.

Convergence is attained even with bigger time steps

(20 times bigger).