Post on 13-Jan-2016
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)
• 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
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
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
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
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
xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 xi+3/2 x
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
1 dtwdtwxxdtNwdttxf n
iiiniii
nii
t
t i
n
n
1st order
2)(),(
2
1211112/1112/1
1 dtwdtwxxdtNwdttxf n
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
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
1 dtwdtwxxdtNwdttxf n
iiiniii
nii
t
t i
n
n
1st order
2)(),(
2
1211112/1112/1
1 dtwdtwxxdtNwdttxf n
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
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
1 dtwdtwxxdtNwdttxf n
iiiniii
nii
t
t i
n
n
1st order
2)(),(
2
1211112/1112/1
1 dtwdtwxxdtNwdttxf n
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 )(),(~
xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 xi+3/2 x
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
• 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
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
After one advective period
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
xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 xi+3/2 x
Control Volume
Ni-1
Ni+1
Ni
Ni+1/2
tn
NU
NC
NDNF
High order solution to be limited
After one advective period
Fromm method associated with the universal limiter
• 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
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
xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 xi+3/2 x
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
• 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
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
Celerity depending on the x axisCelerity
x
ii
i
wtxw
txNtxN
),(
),(~
),(
2/1
over ),(
,12/12/1
nnii
tt
xx
Celerity depending on the x axisCelerity
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
• 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
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
Positive streamer propagation
x=0
ZoomCharge density (C)2ns
x=1cm
UPWIND
Positive streamer propagation
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