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Numerical Schemes for Streamer Discharges at Atmospheric Pressure

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

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Numerical Schemes for Streamer Discharges at Atmospheric Pressure

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

  12. Advect exactly tn+1 tn xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 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. Approximated slopes Upwind * Beam-Warming ** Fromm ** Lax-Wendroff ** ** Second order accurate * First order accurate xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 xi+3/2 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 ● 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 tn Ni+1 Ni+1/2 ND Ni NF Ni-1 NC NU xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 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. tn+1 tn xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 xi+3/2 x 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 Yi+1 Yi Yi-1 tn Yi* Yi-2 dt.wi xi-2 xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 xi+3/2 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 yi+1 2 Lax-Wendroff 2nd order 3 yi-2yi-1 yi yi+1 QUICKEST 3 (Leonard) 3rd order 5 yi-3 yi-2yi-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-2 xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 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 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 E=52kV/cm radius = 200µm ncell=1200 x=1cm x=0 1014cm-3 Initial electron density 108cm-3 x=1cm x=0 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 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

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