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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

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

- 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

2D schemes for discharge simulation

real 2D schemes

2D = 1D + 1D (splitting)

Coupled continuity equations

Poisson equation

S’ can be calculated apart (RK)

and

- 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

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

and

Integration over the control volume :

Introducing a cell average of N(x,t):

then :

and

Integration over the control volume :

Introducing a cell average of N(x,t):

then :

and

Integration over the control volume :

Introducing a cell average of N(x,t):

then :

How to compute

?

over

Assuming that :

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

tn+1

tn

xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 xi+3/2

x

1st order

1st order

Note that : if

and

1st order

Note that : if

and

UPWIND scheme

1st order

Note that : if

and

UPWIND scheme

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

ntotal = 401

w

Periodic boundary conditions

Lax-Wendroff

Upwind

Fromm

Beam-Warming

- 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

f : correction factor

Smoothness indicator near

the right interface of the cell

How to find limiters ?

● 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

● 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

minmod

superbee

Woodward

Van Leer

minmod

Van Leer

Woodward

superbee

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

Fromm method associated with the universal limiter

- 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

tn+1

tn

xi-3/2 xi-1 xi-1/2 xi xi+1/2 xi+1 xi+3/2

x

Finite Volume Discretization

Assuming that y is known :

- 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)

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

……

……

……

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

- 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

● Ncell = 401, after 5 periods

● Ncell = 401, after 500 periods

MUSCL superbee MUSCL Woodward

QUICKEST 3 QUICKEST 5

MUSCL superbee MUSCL Woodward QUICKEST 3 QUICKEST 5

Celerity

x

over

Celerity

x

over

Celerity

x

over

Quickest 5

Quickest 3

After 500 periods

Woodward

Initial profile

x

- 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

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

Charge density (C)

2ns

Zoom

UPWIND

x=0

x=1cm

Charge density (C)

2ns

Zoom

UPWIND

x=0

x=1cm

Charge density (C)

4ns

Quickest

Woodward

Zoom

superbee

minmod

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