Numerical Schemes for Streamer Discharges at Atmospheric Pressure Jean PAILLOL*, Delphine BESSIERES - University of Pau Anne BOURDON – CNRS EM2C Centrale

Numerical Schemes for Streamer Discharges at Atmospheric Pressure

Jean PAILLOL*, Delphine BESSIERES - University of PauAnne BOURDON – CNRS EM2C Centrale ParisPierre SEGUR – CNRS CPAT University of ToulouseArmelle MICHAU, Kahlid HASSOUNI - CNRS LIMHP Paris XIIIEmmanuel MARODE – CNRS LPGP Paris XI

STREAMER GROUP

The Multiscale Nature of Spark Precursors and High Altitude LightningWorkshop May 9-13 – Leiden University - Nederland

• Plasma equations• Integration – Finite Volume Method• Advection by second order schemes• Limiters – TVD – Universal Limiter• Higher order schemes – 3 and 5 – Quickest• Numerical tests – advection• Numerical tests – positive streamer• Conclusion

Outline

Equations in one spatial dimension

1)()( peee

eeee NNWNSx

ND

xx

WN

t

N

21

)( pnpeee

ppp NNNNWNSx

WN

t

N

2

)( pneennn NNWN

x

WN

t

N

)(0

nep NNNe

Ediv

Coupled continuity equations

Poisson equation

real 2D schemes 2D = 1D + 1D (splitting)

2D schemes for discharge simulation

Advection equation – 1D

peeeeeee NNWNSx

ND

xx

WN

t

N

)()(

')(S

x

WN

t

N eee

0)(

x

WN

t

N eee

0)(

x

wN

t

N

0)),((),(

x

txf

t

txN ),(),(),( txNtxwtxf and

S’ can be calculated apart (RK)


Outline

Finite Volume Discretization

i-2 i-1 i i+1 i+2

n-1

n

n+1

t

x

Computational cells

Control Volume

i-3/2 i-1/2 i+1/2 i+3/2

UPWIND

Integration

0),(),(),( 2/12/1

2/1

2/1

txftxfdxtxNdt

dii

x

x

i

i

0)),((),(

x

txf

t

txN),(),(),( txNtxwtxf

Integration over the control volume :

then :

11

),(),(1

)()( 2/12/12/12/1

1n

n

n

n

t

t i

t

t iii

ni

ni dttxfdttxf

xxtNtN

2/1

2/1

),(1

)(2/12/1

i

i

x

xii

i dxtxNxx

tN

Introducing a cell average of N(x,t):

and

Integration

0),(),(),( 2/12/1

2/1

2/1

txftxfdxtxNdt

dii

x

x

i

i

0)),((),(

x

txf

t



then :

11

),(),(1

)()( 2/12/12/12/1

1n

n

n

n

t

t i

t

t iii

ni

ni dttxfdttxf

xxtNtN

2/1

2/1

),(1

)(2/12/1

i

i

x

xii

i dxtxNxx

tN


and

Integration

0),(),(),( 2/12/1

2/1

2/1

txftxfdxtxNdt

dii

x

x

i

i

0)),((),(

x

txf

t



then :

11

),(),(1

)()( 2/12/12/12/1

1n

n

n

n

t

t i

t

t iii

ni

ni dttxfdttxf

xxtNtN

2/1

2/1

),(1

)(2/12/1

i

i

x

xii

i dxtxNxx

tN


and

Flux approximation

1

),( 2/1

n

n

t

t i dttxf

),(),(),( 2/12/12/1 txNtxwtxf iii

How to compute ?

Assuming that :

),(~

),( 2/12/1 txNwtxf iiii

ii

i

wtxw

txNtxN

),(

),(~

),(

2/1

over

),(

,12/12/1

nnii

tt

xx

Flux approximation

ni

ni NtxN ),(~

nii

ni

ni xxNtxN )(),(~

0th order

1st order

xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 xi+3/2 x

Control Volume

How to choose the approximated value ?),(~

2/1 txN ii

Linear approximation

Advect exactly

2)(),(

22

2/12/1

1 dtwdtwxxdtNwdttxf n

iiiniii

nii

t

t i

n

n


tn

tn+1

iw

1st order

11

),(~

),( 2/12/1

n

n

n

n

t

t iii

t

t i dttxNwdttxf

2/1

2/1

)(~i

ii

x

dtwx

ni dxxN

1

))((~

2/1

n

n

t

t

nii

nii dtttwxNw

Update averages [LeVeque]

11

),(1

),(1

)()( 2/12/11

n

n

n

n

t

t i

t

ti

ii

ni

ni dttxf

dxdttxf

dxtNtN

wwi dxdxi

2)(),(

22

2/12/1


iiiniii

nii

t

t i

n

n

1st order

2)(),(

2

1211112/1112/1


iiiniii

nii

t

t i

n

n

Note that : if

)(22

)(1

1

22

11 n

ini

ni

ni

ni

ni

dtwdt

dxwNNwdt

dxNN

and


11

),(1

),(1

)()( 2/12/11

n

n

n

n

t

t i

t

ti

ii

ni

ni dttxf

dxdttxf

dxtNtN

wwi dxdxi

2)(),(

22

2/12/1


iiiniii

nii

t

t i

n

n

1st order

2)(),(

2

1211112/1112/1


iiiniii

nii

t

t i

n

n

Note that : if

)(22

)(1

1

22

11 n

ini

ni

ni

ni

ni

dtwdt

dxwNNwdt

dxNN

and

UPWIND scheme


11

),(1

),(1

)()( 2/12/11

n

n

n

n

t

t i

t

ti

ii

ni

ni dttxf

dxdttxf

dxtNtN

wwi dxdxi

2)(),(

22

2/12/1


iiiniii

nii

t

t i

n

n

1st order

2)(),(

2

1211112/1112/1


iiiniii

nii

t

t i

n

n

Note that : if

)(22

)(1

1

22

11 n

ini

ni

ni

ni

ni

dtwdt

dxwNNwdt

dxNN

and

UPWIND scheme

Approximated slopes

0ni

ii

ni

nin

i xx

NN

1

1

1

1

ii

ni

nin

i xx

NN

nii

ni

ni xxNtxN )(),(~


11

11

ii

ni

nin

i xx

NN

Upwind *

Lax-Wendroff **

Beam-Warming **

Fromm **

* First order accurate ** Second order accurate

Numerical experiments [Toro]

Periodic boundary conditions

ntotal = 401

w

4.0dx

dtw

After one advective period

Upwind Lax-Wendroff

Beam-Warming Fromm


Outline

Slope Limiters

)(22

)(1

1

22

11 n

ini

ni

ni

ni

ni

dtwdt

dxwNNwdt

dxNN

)(12

)( 111 n

ini

ni

ni

ni

ni dx

wdtwdtNN

dx

wdtNN

))(( 12/1ni

ni

ni

ni NN

ni

ni

ni

nin

i NN

NN

1

12/1

How to find limiters ?

Smoothness indicator nearthe right interface of the cell

: correction factor

TVD Methods

)()( 1 nn NTVNTV

● Motivation

First order schemes poor resolution, entropy satisfying and non oscillatory solutions.Higher order schemes oscillatory solutions at discontinuities.

● Good criterion to design “high order” oscillation free schemes is based on the Total Variation of the solution.

● Total Variation of the discrete solution :

● Total Variation of the exact solution is non-increasing TVD schemes

i

ni

ni

n NNNTV 1)(

Total Variation Diminishing Schemes

TVD Methods

0)2,2mod(min)(0

● Godunov’s theorem : No second or higher order accurate constant coefficient (linear) scheme can be TVD higher order TVD schemes must be nonlinear.

● Harten’s theorem :

TVD region

Fromm

gWarBeam

WendroffLax

upwind

2

1)(

min)(

1)(

0)(

TVD Methods

● Sweby’s suggestion :

2nd order

Avoid excessive compression of solutions

2nd order

Second order TVD schemes

1)(

))2,2,2

1min(,0max()(

)),2min(),2,1min(,0max()(

),1mod(min)( minmod

superbee

Woodward

Van Leer


minmod Van Leer

Woodward superbee

Universal Limiter [Leonard]

nHiN 2/1

nHi

i

inHi

i

ini

ni N

dx

dtwN

dx

dtwNN 2/1

1

2/12/1

2/11


Control Volume

Ni-1

Ni+1

Ni

Ni+1/2

tn

NU

NC

NDNF

High order solution to be limited


Fromm method associated with the universal limiter


Outline

Advect exactly

11

),(),(1

)()( 2/12/11

n

n

n

n

t

t i

t

t ii

ni

ni dttxfdttxf

dxtNtN

11

),(),( 2/12/1

n

n

n

n

t

t ii

t

t i dttxNwdttxf

2/1

2/1

)(i

ii

x

dtwx

n dxxN

Finite Volume Discretization


tn

tn+1

iw

Integration [Leonard]

dx

xdxN

)()(

2/1

2/12/1

11 )(1),(

1),(

12/12/1

i

ii

n

n

n

n

x

dtwxi

t

t iii

t

t ii

dxdx

xd

dxdttxN

dxdttxf

dx

Assuming that is known :

i

ii

i

dtwxi

dxdxii

*2/12/1

1

111 **)()(

i

ii

i

iini

ni dxdx

tNtN

High order approximation of *

xi-2 xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 xi+3/2 x

Control Volume

i

tn

i+1

i-1

i-2

i*

dt.wi

Polynomial interpolation of (x) i*

function is determined at the boundaries of the control cell by numerical integration

High order approximation of *

* is determined by polynomial interpolation

Polynomial order Interpolation points Numerical scheme

1

2

i-1 i UPWIND

i-1i i+1 Lax-Wendroff2nd order

3 i-2 i-1i i+1 QUICKEST 3 (Leonard)3rd order

5 QUICKEST 5 (Leonard)5th order

…… …… ……

i-3 i-2 i-1i i+1 i+2

Universal Limiter applied to * [Leonard]

(x) is a continuously increasing function (monotone)

xi-2 xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 xi+3/2 x

i

tn

i+1

i-1i-2

i*

dt.wi


Outline

Numerical advection tests

MUSCL superbee MUSCL WoodwardQUICKEST 3 QUICKEST 5

● Ncell = 401, after 5 periods

● Ncell = 401, after 500 periods

Ncell = 1601, after 500 periods

MUSCL superbee MUSCL Woodward QUICKEST 3 QUICKEST 5

Celerity depending on the x axisCelerity

x

ii

i

wtxw

txNtxN

),(

),(~

),(

2/1

over ),(

,12/12/1

nnii

tt

xx


x

ii

i

wtxw

txNtxN

),(

),(~

),(

2/1

over ),(

,12/12/1

nnii

tt

xx


x

Initial profile

Quickest 5

Quickest 3

Woodward

x

After 500 periods

ii

i

wtxw

txNtxN

),(

),(~

),(

2/1

over ),(

,12/12/1

nnii

tt

xx


Outline

Positive streamer propagation

Cathode Anode

x=0 x=1cm

x=0 x=1cm

Plan to plan electrode system [Dahli and Williams]

Initial electron density

108cm-3

1014cm-3

x=0.9cm

E=52kV/cmradius = 200µmncell=1200

streamer


x=0

ZoomCharge density (C)2ns

x=1cm

UPWIND


x=0

ZoomCharge density (C)2ns

Charge density (C)4ns

x=1cm

Zoom

superbeeminmod

Woodward Quickest

UPWIND

Conclusion

Is it worth working on accurate scheme for streamer modelling ?

YES !

especially in 2D numerical simulations

Quickest 5

Quickest 3

TVD minmod

Error (%)0.783.8

3.4126.522.77

Number of cells1601401

1601201

1601

Advection tests

High order schemes may be useful

Documents

Numerical Schemes for Streamer Discharges at Atmospheric Pressure Jean PAILLOL*, Delphine BESSIERES - University of Pau Anne BOURDON – CNRS EM2C Centrale