Numerical Schemes for
Streamer Discharges at
Atmospheric Pressure
Jean PAILLOL*, Delphine BESSIERES - University of Pau
Anne BOURDON – CNRS EM2C Centrale Paris
Pierre SEGUR – CNRS CPAT University of Toulouse
Armelle MICHAU, Kahlid HASSOUNI - CNRS LIMHP Paris XIII
Emmanuel MARODE – CNRS LPGP Paris XI
STREAMER GROUP
The Multiscale Nature of Spark Precursors and High Altitude Lightning
Workshop May 9-13 – Leiden University - Nederland
Outline
•
•
•
•
•
•
•
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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
Equations in one spatial dimension
2D schemes for discharge simulation
real 2D schemes
2D = 1D + 1D (splitting)
Coupled continuity equations
N e  ( N eWe )   N e 

 D
  S  (   ) N e We  N e N p 1
t
x
x  x 
N p ( N pW p )

 S  N e We  N e N p 1  N n N p  2
t
x
N n  ( N nWn )

 N e We  N n N p  2
t
x
Poisson equation

e
div E   ( N p  N e  N n )
0
Advection equation – 1D
N e  ( N eWe )   N e 

 D
  S  (   ) N e We  N e N p 
t
x
x  x 
N e  ( N eWe )

 S'
t
x
N e  ( N eWe )

0
t
x
S’ can be calculated apart (RK)
N  ( wN )

0
t
x
N ( x, t )  ( f ( x, t ))

 0 and
t
x
f ( x, t )  w( x, t ) N ( x, t )
Outline
•
•
•
•
•
•
•
•
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
Finite Volume Discretization
Computational cells
t
n+1
UPWIND
n
n-1
i-2
i-1
i-3/2
i
i-1/2
i+1
i+1/2
Control Volume
i+2
i+3/2
x
Integration
N ( x, t )  ( f ( x, t ))

0
t
x
and
f ( x, t )  w( x, t ) N ( x, t )
Integration over the control volume :
d xi1 / 2
N ( x, t )dx  f ( xi 1/ 2 , t )  f ( xi 1/ 2 , t )  0

dt xi1 / 2
Introducing a cell average of N(x,t):
1
N i (t ) 
xi 1/ 2  xi 1/ 2

xi 1 / 2
xi 1 / 2
N ( x, t )dx
then :
N i (t
n 1
1
)  N i (t ) 
xi 1/ 2  xi 1/ 2
n
 t f ( x , t )dt  t f ( x , t )dt 
i 1 / 2
i 1 / 2
t n
 t n

n1
n1
Integration
N ( x, t )  ( f ( x, t ))

0
t
x
and
f ( x, t )  w( x, t ) N ( x, t )
Integration over the control volume :
d xi1 / 2
N ( x, t )dx  f ( xi 1/ 2 , t )  f ( xi 1/ 2 , t )  0

dt xi1 / 2
Introducing a cell average of N(x,t):
1
N i (t ) 
xi 1/ 2  xi 1/ 2

xi 1 / 2
xi 1 / 2
N ( x, t )dx
then :
N i (t
n 1
1
)  N i (t ) 
xi 1/ 2  xi 1/ 2
n
 t f ( x , t )dt  t f ( x , t )dt 
i 1 / 2
i 1 / 2
t n
 t n

n1
n1
Integration
N ( x, t )  ( f ( x, t ))

0
t
x
and
f ( x, t )  w( x, t ) N ( x, t )
Integration over the control volume :
d xi1 / 2
N ( x, t )dx  f ( xi 1/ 2 , t )  f ( xi 1/ 2 , t )  0

dt xi1 / 2
Introducing a cell average of N(x,t):
1
N i (t ) 
xi 1/ 2  xi 1/ 2

xi 1 / 2
xi 1 / 2
N ( x, t )dx
then :
N i (t
n 1
1
)  N i (t ) 
xi 1/ 2  xi 1/ 2
n
 t f ( x , t )dt  t f ( x , t )dt 
i 1 / 2
i 1 / 2
t n
 t n

n1
n1
Flux approximation
How to compute

t n1
t
n
f ( xi 1/ 2 , t )dt
?
f ( xi 1/ 2 , t )  w( xi 1/ 2 , t ) N ( xi 1/ 2 , t )
~
N ( x , t )  N i ( x, t )
Assuming that :
w( xi 1/ 2 , t )  wi
over
~
f ( xi 1/ 2 , t )  wi Ni ( xi 1/ 2 , t )
xi 1/ 2 , xi 1/ 2 
(t n , t n 1 )
Flux approximation
How to choose the approximated value
0th order
1st order
xi-3/2
~
Ni ( xi 1/ 2 , t )
~
n
Ni ( x, t n )  Ni
~
Ni ( x, t n )  Nin  ( x  xi ) in
xi-1
xi-1/2
xi
xi+1/2
Control Volume
?
Linear approximation
xi+1
xi+3/2
x
Advect exactly
tn+1
wi
tn
xi-3/2

t n1
tn
xi-1
xi-1/2
xi
t n1
f ( xi 1/ 2 , t )dt  wi  n
t
t n1
 wi  n
t
1st

order
t n1
tn

xi+1/2
xi+1
xi+3/2 x
~
N i ( xi 1/ 2 , t )dt
~n
N i ( xi 1/ 2  wi (t  t n )) dt
xi 1 / 2
xi 1 / 2  wi dt
~n
N i ( x)dx
2
dt
f ( xi 1/ 2 , t )dt  wi N in dt  ( xi 1/ 2  xi ) in wi dt  wi2 in
2
Update averages [LeVeque]
1
n 1
n
N i (t )  N i (t ) 
dxi

t n1
tn
1 t n1
f ( xi 1/ 2 , t )dt 
f ( xi 1/ 2 , t )dt
n

t
dxi
1st order


t n1
tn
t n1
tn
2
dt
f ( xi 1/ 2 , t )dt  wi N in dt  ( xi 1/ 2  xi ) in wi dt  wi2 in
2
2
dt
f ( xi 1/ 2 , t )dt  wi 1 N in1dt  ( xi 1/ 2  xi 1 ) in1wi 1dt  wi21 in1
2
Note that : if
dxi  dx
and
wi  w
2
2



 n
1
dx
w
dt
n 1
n
n
n
n
( i   i 1 )
N i  N i   wdt ( N i  N i 1 )   w dt 
dx 
2 
 2

Update averages [LeVeque]
1
n 1
n
N i (t )  N i (t ) 
dxi

t n1
tn
1 t n1
f ( xi 1/ 2 , t )dt 
f ( xi 1/ 2 , t )dt
n

t
dxi
1st order


t n1
tn
t n1
tn
2
dt
f ( xi 1/ 2 , t )dt  wi N in dt  ( xi 1/ 2  xi ) in wi dt  wi2 in
2
2
dt
f ( xi 1/ 2 , t )dt  wi 1 N in1dt  ( xi 1/ 2  xi 1 ) in1wi 1dt  wi21 in1
2
Note that : if
dxi  dx
and
wi  w
2
2



 n
1
dx
w
dt
n 1
n
n
n
n
( i   i 1 )
N i  N i   wdt ( N i  N i 1 )   w dt 
dx 
2 
 2

UPWIND scheme
Update averages [LeVeque]
1
n 1
n
N i (t )  N i (t ) 
dxi

t n1
tn
1 t n1
f ( xi 1/ 2 , t )dt 
f ( xi 1/ 2 , t )dt
n

t
dxi
1st order


t n1
tn
t n1
tn
2
dt
f ( xi 1/ 2 , t )dt  wi N in dt  ( xi 1/ 2  xi ) in wi dt  wi2 in
2
2
dt
f ( xi 1/ 2 , t )dt  wi 1 N in1dt  ( xi 1/ 2  xi 1 ) in1wi 1dt  wi21 in1
2
Note that : if
dxi  dx
and
wi  w
2
2



 n
1
dx
w
dt
n 1
n
n
n
n
( i   i 1 )
N i  N i   wdt ( N i  N i 1 )   w dt 
dx 
2 
 2

UPWIND scheme
Approximated slopes
~
Ni ( x, t n )  Nin  ( x  xi ) in
 0
n
i
Upwind *
n
n
N

N
i Lax-Wendroff **
 in  i 1
xi 1  xi
** Second order accurate
* First order accurate
xi-3/2
xi-1
xi-1/2
N in  N in1
Beam-Warming **
 
xi  xi 1
N in1  N in1 Fromm **
n
i 
xi 1  xi 1
n
i
xi
xi+1/2
xi+1
xi+3/2
x
Numerical experiments [Toro]
ntotal = 401
w dt
 0.4
dx
w
Periodic boundary conditions
After one advective period
Upwind
Beam-Warming
Lax-Wendroff
Fromm
Outline
•
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•
•
•
•
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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
Slope Limiters
2
2



 n
1
dx
w
dt
n 1
n
n
n
n
( i   i 1 )
N i  N i   wdt ( N i  N i 1 )   w dt 
dx 
2 
 2

N
n 1
i
wdt
wdt  wdt  n
n
n
n
N 
( N i  N i 1 ) 
1 
( i   i 1 )
dx
2 
dx 
n
i
 in   ( in1/ 2 )( N in1  N in )
 in1/ 2
N in  N in1
 n
n
N i 1  N i
 : correction factor
Smoothness indicator near
the right interface of the cell
How to find limiters ?
TVD Methods
● 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 :
TV ( N n )   N in1  N in
i
● Total Variation of the exact solution is non-increasing  TVD schemes
TV ( N n1 )  TV ( N n )
Total Variation Diminishing Schemes
TVD Methods
● 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 : 0   ( )  min mod( 2,2 )
  0
 ( )  0 upwind
 ( )  1 Lax  Wendroff
 ( )   Beam  Warmin g
 1
 ( ) 
Fromm
2
TVD region
TVD Methods
● Sweby’s suggestion :
2nd order
Avoid excessive compression of solutions
2nd order
Second order TVD schemes
minmod
 ( )  min mod(1,  )
 ( )  max( 0, min( 1,2 ), min( 2,  )) superbee
1
 ( )  max( 0, min(
,2,2 ))
Woodward
2

 ( ) 
1 
Van Leer
After one advective period
minmod
Van Leer
Woodward
superbee
Universal Limiter [Leonard]
N
n 1
i
wi 1/ 2 dt H n wi 1/ 2 dt H n
N 
N i 1/ 2 
N i 1/ 2
dxi
dxi 1
n
i
N iH1n/ 2
High order solution to be limited
tn
Ni+1
Ni+1/2
Ni
NF
Ni-1
ND
NC
NU
xi-3/2
xi-1
xi-1/2
xi
xi+1/2
Control Volume
xi+1
xi+3/2
x
After one advective period
Fromm method associated with the universal limiter
Outline
•
•
•
•
•
•
•
•
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
Advect exactly
Finite Volume Discretization
1
N i (t n 1 )  N i (t n ) 
dxi

t n1
t
n
 t f ( x , t )dt  t f ( x , t )dt 
i 1 / 2
i 1 / 2
t n
 t n

n 1
n 1
t n1
f ( xi 1/ 2 , t )dt  wi  n N ( xi 1/ 2 , t )dt
t

xi 1 / 2
xi 1 / 2  wi dt
N n ( x)dx
tn+1
wi
tn
xi-3/2
xi-1
xi-1/2
xi
xi+1/2
xi+1
xi+3/2x
Integration [Leonard]
d ( x )
N ( x) 
dx
Assuming that  is known :
1
dxi

t n1
tn
1
f ( xi 1/ 2 , t )dt 
dxi

N i (t
n 1

t n1
tn
1
N i ( xi 1/ 2 , t )dt 
dxi
 i  x
i 1 / 2  wi 1 / 2 dt
dxi
)  N i (t ) 
n

d ( x)
xi1/ 2 wi1/ 2dt dx dx
xi 1 / 2
 i  i *
dxi
 i  i *  i 1  i 1 *
dxi

dxi 1
High order approximation of *
Y function is determined at the boundaries of the control cell
by numerical integration
Yi+1
Yi
Yi-1
tn
Y i*
Yi-2
dt.wi
xi-2
xi+3/2
xi-3/2
xi-1
xi-1/2
xi
xi+1/2
xi+1
Control Volume
Polynomial interpolation of (x)
Y i*
x
High order approximation of *
* is determined by polynomial interpolation
Polynomial order
Interpolation points
Numerical scheme
1
i-1 i
UPWIND
2
i-1 i i+1
3
i-2 i-1 i i+1
5
i-3 i-2 i-1 i i+1 i+2
……
……
Lax-Wendroff
2nd order
QUICKEST 3 (Leonard)
3rd order
QUICKEST 5 (Leonard)
5th order
……
Universal Limiter applied to * [Leonard]
(x) is a continuously increasing function (monotone)
Yi+1
dt.wi
Y i*
xi-3/2
Yi
Yi-1
Yi-2
xi-2
xi+3/2
tn
xi-1
xi-1/2
xi
xi+1/2
xi+1
x
Outline
•
•
•
•
•
•
•
•
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
Numerical advection tests
● Ncell = 401, after 5 periods
● Ncell = 401, after 500 periods
MUSCL superbee
QUICKEST 3
MUSCL Woodward
QUICKEST 5
Ncell = 1601, after 500 periods
MUSCL superbee
MUSCL Woodward
QUICKEST 3
QUICKEST 5
Celerity depending on the x axis
~
N ( x , t )  N i ( x, t )
Celerity
w( xi 1/ 2 , t )  wi
over
xi 1/ 2 , xi 1/ 2 
(t n , t n 1 )
x
Celerity depending on the x axis
~
N ( x , t )  N i ( x, t )
Celerity
w( xi 1/ 2 , t )  wi
over
xi 1/ 2 , xi 1/ 2 
(t n , t n 1 )
x
Celerity depending on the x axis
~
N ( x , t )  N i ( x, t )
Celerity
w( xi 1/ 2 , t )  wi
over
x
xi 1/ 2 , xi 1/ 2 
(t n , t n 1 )
Quickest 5
Quickest 3
After 500 periods
Woodward
Initial profile
x
Outline
•
•
•
•
•
•
•
•
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
Positive streamer propagation
Plan to plan electrode system [Dahli and Williams]
streamer
Cathode
x=0
Initial electron density
Anode
x=1cm
1014cm-3
108cm-3
x=0
x=1cm
x=0.9cm
E=52kV/cm
radius = 200µm
ncell=1200
Positive streamer propagation
Charge density (C)
2ns
x=0
Zoom
UPWIND
x=1cm
Positive streamer propagation
Charge density (C)
2ns
x=0
Zoom
UPWIND
x=1cm
Charge density (C)
4ns
Woodward Quickest
minmod
superbee
Zoom
Conclusion
Is it worth working on accurate scheme for streamer modelling ?
YES !
especially in 2D numerical simulations
Advection tests
Quickest 5
Quickest 3
TVD minmod
Error (%)
0.78
3.8
3.41
26.5
22.77
Number of cells
1601
401
1601
201
1601
High order schemes may be useful