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 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 Poisson equation

Advection equation – 1D S’ can be calculated apart (RK) and

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 x i-2 i-1 i i+1 i+2 i-3/2 i-1/2 i+1/2 i+3/2 Control Volume

Integration and Integration over the control volume : Introducing a cell average of N(x,t): then :

Integration and Integration over the control volume : Introducing a cell average of N(x,t): then :

Integration and Integration over the control volume : Introducing a cell average of N(x,t): then :

Flux approximation How to compute ? over Assuming that :

How to choose the approximated value ? 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

Advect exactly x tn+1 tn 1st order xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 xi+3/2 x 1st order

Update averages [LeVeque] 1st order Note that : if and

Update averages [LeVeque] 1st order Note that : if and UPWIND scheme

Update averages [LeVeque] 1st order Note that : if and UPWIND scheme

** Second order accurate Approximated slopes Upwind * Beam-Warming ** Lax-Wendroff ** Fromm ** * First order accurate ** Second order accurate xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 xi+3/2 x

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

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

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

Slope Limiters f : correction factor How to find limiters ? Smoothness indicator near the right interface of the cell How to find limiters ?

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

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

TVD Methods ● Sweby’s suggestion : 2nd order Avoid excessive compression of solutions 2nd order

Second order TVD schemes minmod superbee Woodward Van Leer

After one advective period minmod Van Leer Woodward superbee

Universal Limiter [Leonard] High order solution to be limited Ni+1 tn Ni+1/2 Ni ND 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

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 x Finite Volume Discretization tn+1 tn xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 xi+3/2 x tn tn+1

Integration [Leonard] Assuming that y is known :

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 Polynomial interpolation of y(x) Yi*

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 …… …… ……

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

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 MUSCL Woodward QUICKEST 3 QUICKEST 5

Ncell = 1601, after 500 periods MUSCL superbee MUSCL Woodward QUICKEST 3 QUICKEST 5

Celerity depending on the x axis over

Celerity depending on the x axis over

Celerity depending on the x axis over 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 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

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

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

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