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

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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


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 :


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

tn+1

tn

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


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


Numerical experiments [Toro]

ntotal = 401

w

Periodic boundary conditions


After one advective period

Lax-Wendroff

Upwind

Fromm

Beam-Warming


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

Smoothness indicator near

the right interface of the cell

How to find limiters ?


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

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


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


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


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

Yi*

Polynomial interpolation of y(x)


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

……

……

……


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

Celerity

x

over


Celerity depending on the x axis

Celerity

x

over


Celerity depending on the x axis

Celerity

x

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=1cm

x=0

1014cm-3

Initial electron density

108cm-3

x=1cm

x=0

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

Quickest

Woodward

Zoom

superbee

minmod


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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