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Numerical Schemes for Streamer Discharges at Atmospheric Pressure

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

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Title: Numerical Schemes for Streamer Discharges at Atmospheric Pressure


1
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
2
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

3
Equations in one spatial dimension
2D schemes for discharge simulation
real 2D schemes
2D 1D 1D (splitting)
Coupled continuity equations
Poisson equation
4
Advection equation 1D
S can be calculated apart (RK)
and
5
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

6
Finite Volume Discretization
Computational cells
t
n1
UPWIND
n
n-1
x
i-2 i-1 i
i1 i2
i-3/2 i-1/2 i1/2 i3/2
Control Volume
7
Integration
and
Integration over the control volume
Introducing a cell average of N(x,t)
then
8
Integration
and
Integration over the control volume
Introducing a cell average of N(x,t)
then
9
Integration
and
Integration over the control volume
Introducing a cell average of N(x,t)
then
10
Flux approximation
How to compute
?
over
Assuming that
11
Flux approximation
How to choose the approximated value
?
0th order
1st order
Linear approximation
xi-3/2 xi-1 xi-1/2 xi
xi1/2 xi1 xi3/2
x
Control Volume
12
Advect exactly
tn1
tn
xi-3/2 xi-1 xi-1/2 xi
xi1/2 xi1 xi3/2
x
1st order
13
Update averages LeVeque
1st order
Note that if
and
14
Update averages LeVeque
1st order
Note that if
and
UPWIND scheme
15
Update averages LeVeque
1st order
Note that if
and
UPWIND scheme
16
Approximated slopes
Upwind
Beam-Warming
Fromm
Lax-Wendroff
Second order accurate
First order accurate
xi-3/2 xi-1 xi-1/2 xi
xi1/2 xi1 xi3/2
x
17
Numerical experiments Toro
ntotal 401
w
Periodic boundary conditions
18
After one advective period
Lax-Wendroff
Upwind
Fromm
Beam-Warming
19
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

20
Slope Limiters
f correction factor
Smoothness indicator near the right interface of
the cell
How to find limiters ?
21
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
? Total Variation of the exact solution is
non-increasing ? TVD schemes
Total Variation Diminishing Schemes
22
TVD Methods
? Godunovs theorem No second or higher order
accurate constant coefficient (linear) scheme
can be TVD ? higher order TVD schemes must be
nonlinear.
? Hartens theorem
TVD region
23
TVD Methods
? Swebys suggestion
2nd order
Avoid excessive compression of solutions
2nd order
24
Second order TVD schemes
minmod
superbee
Woodward
Van Leer
25
After one advective period
minmod
Van Leer
Woodward
superbee
26
Universal Limiter Leonard
High order solution to be limited
tn
Ni1
Ni1/2
ND
Ni
NF
Ni-1
NC
NU
xi-3/2 xi-1 xi-1/2 xi
xi1/2 xi1 xi3/2
x
Control Volume
27
After one advective period
Fromm method associated with the universal limiter
28
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

29
Advect exactly
Finite Volume Discretization
30
Integration Leonard
Assuming that y is known
31
High order approximation of y
  • function is determined at the boundaries of the
    control cell
  • by numerical integration

Yi1
Yi
Yi-1
tn
Yi
Yi-2
dt.wi
xi-2 xi-3/2 xi-1 xi-1/2
xi xi1/2 xi1 xi3/2
x
Control Volume
Yi
Polynomial interpolation of y(x)
32
High order approximation of y
y is determined by polynomial interpolation
Polynomial order
Interpolation points
Numerical scheme
yi-1 yi
UPWIND
1
yi-1 yi yi1
2
Lax-Wendroff 2nd order
3
yi-2 yi-1 yi yi1
QUICKEST 3 (Leonard) 3rd order
5
yi-3 yi-2 yi-1 yi yi1 yi2
QUICKEST 5 (Leonard) 5th order



33
Universal Limiter applied to y Leonard
y(x) is a continuously increasing function
(monotone)
Yi1
dt.wi
tn
Yi
Yi
Yi-1
Yi-2
xi-2 xi-3/2 xi-1 xi-1/2
xi xi1/2 xi1 xi3/2
x
34
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

35
Numerical advection tests
? Ncell 401, after 5 periods
? Ncell 401, after 500 periods
MUSCL superbee MUSCL Woodward QUICKEST 3
QUICKEST 5
36
Ncell 1601, after 500 periods
MUSCL superbee MUSCL Woodward
QUICKEST 3 QUICKEST 5
37
Celerity depending on the x axis
Celerity
x
over
38
Celerity depending on the x axis
Celerity
x
over
39
Celerity depending on the x axis
Celerity
x
over
Quickest 5
Quickest 3
After 500 periods
Woodward
Initial profile
x
40
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

41
Positive streamer propagation
Plan to plan electrode system Dahli and
Williams
streamer
Cathode
Anode
E52kV/cm radius 200µm ncell1200
x1cm
x0
1014cm-3
Initial electron density
108cm-3
x1cm
x0
x0.9cm
42
Positive streamer propagation
Charge density (C) 2ns
Zoom
UPWIND
x0
x1cm
43
Positive streamer propagation
Charge density (C) 2ns
Zoom
UPWIND
x0
x1cm
Charge density (C) 4ns
Quickest
Woodward
Zoom
superbee
minmod
44
Conclusion
Is it worth working on accurate scheme for
streamer modelling ?
YES !
especially in 2D numerical simulations
Advection tests
Error () 0.78 3.8 3.41 26.5 22.77
Number of cells 1601 401 1601 201 1601
Quickest 5 Quickest 3 TVD minmod
High order schemes may be useful
45
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