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
• 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
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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

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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
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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

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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
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Integration
and
Integration over the control volume
Introducing a cell average of N(x,t)
then
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Integration
and
Integration over the control volume
Introducing a cell average of N(x,t)
then
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Integration
and
Integration over the control volume
Introducing a cell average of N(x,t)
then
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Flux approximation
How to compute
?
over
Assuming that
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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
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Advect exactly
tn1
tn
xi-3/2 xi-1 xi-1/2 xi
xi1/2 xi1 xi3/2
x
1st order
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Update averages LeVeque
1st order
Note that if
and
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Update averages LeVeque
1st order
Note that if
and
UPWIND scheme
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Update averages LeVeque
1st order
Note that if
and
UPWIND scheme
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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
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Numerical experiments Toro
ntotal 401
w
Periodic boundary conditions
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After one advective period
Lax-Wendroff
Upwind
Fromm
Beam-Warming
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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

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Slope Limiters
f correction factor
Smoothness indicator near the right interface of
the cell
How to find limiters ?
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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
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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
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TVD Methods
? Swebys suggestion
2nd order
Avoid excessive compression of solutions
2nd order
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Second order TVD schemes
minmod
superbee
Woodward
Van Leer
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After one advective period
minmod
Van Leer
Woodward
superbee
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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
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After one advective period
Fromm method associated with the universal limiter
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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

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Advect exactly
Finite Volume Discretization
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Integration Leonard
Assuming that y is known
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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)
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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



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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
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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

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

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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
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Positive streamer propagation
Charge density (C) 2ns
Zoom
UPWIND
x0
x1cm
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Positive streamer propagation
Charge density (C) 2ns
Zoom
UPWIND
x0
x1cm
Charge density (C) 4ns
Quickest
Woodward
Zoom
superbee
minmod
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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
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