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
n+1
UPWIND n
n-1 x
i i i i i+2
i-3/ i-1/ i+1/ i+3/2
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. How to choose the approximated value ?
Flux approximation
How to choose the approximated value ?
0th order
1st order
Linear approximation
xi-3/ xi xi-1/ xi xi+1/ xi xi+3/2 x
Control Volume

12. Advect exactly x tn+1 tn 1st order
xi-3/ xi xi-1/ xi xi+1/ xi xi+3/2 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. ** Second order accurate
Approximated slopes
Upwind *
Beam-Warming **
Lax-Wendroff **
Fromm **
* First order accurate
** Second order accurate
xi-3/ xi xi-1/ xi xi+1/ xi xi+3/2 x

17. Numerical experiments [Toro]
ntotal = 401 w
Periodic boundary conditions

18. After one advective period
Upwind
Lax-Wendroff
Beam-Warming
Fromm

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 How to find limiters ?
Smoothness indicator near
the right interface of the cell
How to find limiters ?

21. Total Variation Diminishing Schemes
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
● 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

23. TVD Methods ● Sweby’s 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
Ni+1 tn
Ni+1/2 Ni ND NF
Ni-1 NC NU
xi-3/ xi xi-1/ xi xi+1/ xi xi+3/2 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 x Finite Volume Discretization tn+1 tn
xi-3/ xi xi-1/ xi xi+1/ xi xi+3/2 x tn
tn+1

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
Yi+1 Yi
Yi-1 tn
Yi*
Yi-2
dt.wi
xi xi-3/ xi xi-1/ xi xi+1/ xi xi+3/2 x
Control Volume
Polynomial interpolation of y(x)
Yi*

32. High order approximation of y*
y* is determined by polynomial interpolation
Polynomial order
Interpolation points
Numerical scheme 1
yi-1 yi
UPWIND 2
yi-1 yi yi+1
Lax-Wendroff
2nd order 3
yi-2 yi-1 yi yi+1
QUICKEST 3 (Leonard)
3rd order 5
yi-3 yi-2 yi-1 yi yi+1 yi+2
QUICKEST 5 (Leonard)
5th order …… …… ……

33. Universal Limiter applied to y* [Leonard]
y(x) is a continuously increasing function (monotone)
Yi+1
dt.wi tn
Yi* Yi
Yi-1
Yi-2
xi xi-3/ xi xi-1/ xi xi+1/ xi xi+3/2 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 QUICKEST 5

36. Ncell = 1601, after 500 periods
MUSCL superbee MUSCL Woodward QUICKEST QUICKEST 5

37. Celerity depending on the x axis
over

38. Celerity depending on the x axis
over

39. Celerity depending on the x axis
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
E=52kV/cm
radius = 200µm
ncell=1200
x=0
x=1cm
1014cm-3
Initial electron density
108cm-3
x=0
x=1cm
x=0.9cm

42. Positive streamer propagation
Charge density (C)
2ns
Zoom
UPWIND
x=0
x=1cm

43. Positive streamer propagation
Charge density (C)
2ns
Zoom
UPWIND
x=0
x=1cm
Charge density (C)
4ns
Woodward
Quickest
Zoom
superbee
minmod

44. High order schemes may be useful
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
201
Quickest 5
Quickest 3
TVD minmod
High order schemes may be useful