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

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

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

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

Advection equation – 1D

S’ can be calculated apart (RK)

and


Outline1

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

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

Integration

and

Integration over the control volume :

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

then :


Integration1

Integration

and

Integration over the control volume :

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

then :


Integration2

Integration

and

Integration over the control volume :

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

then :


Flux approximation

Flux approximation

How to compute

?

over

Assuming that :


Flux approximation1

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

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

Update averages [LeVeque]

1st order

Note that : if

and


Update averages leveque1

Update averages [LeVeque]

1st order

Note that : if

and

UPWIND scheme


Update averages leveque2

Update averages [LeVeque]

1st order

Note that : if

and

UPWIND scheme


Approximated slopes

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

Numerical experiments [Toro]

ntotal = 401

w

Periodic boundary conditions


After one advective period

After one advective period

Lax-Wendroff

Upwind

Fromm

Beam-Warming


Outline2

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

Slope Limiters

f : correction factor

Smoothness indicator near

the right interface of the cell

How to find limiters ?


Tvd methods

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 methods1

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


Numerical schemes for streamer discharges at atmospheric pressure

TVD Methods

● Sweby’s suggestion :

2nd order

Avoid excessive compression of solutions

2nd order


Second order tvd schemes

Second order TVD schemes

minmod

superbee

Woodward

Van Leer


After one advective period1

After one advective period

minmod

Van Leer

Woodward

superbee


Universal limiter leonard

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 period2

After one advective period

Fromm method associated with the universal limiter


Outline3

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 exactly1

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

Integration [Leonard]

Assuming that y is known :


High order approximation of y

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 y1

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

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


Outline4

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

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

Ncell = 1601, after 500 periods

MUSCL superbee MUSCL Woodward QUICKEST 3 QUICKEST 5


Celerity depending on the x axis

Celerity depending on the x axis

Celerity

x

over


Celerity depending on the x axis1

Celerity depending on the x axis

Celerity

x

over


Celerity depending on the x axis2

Celerity depending on the x axis

Celerity

x

over

Quickest 5

Quickest 3

After 500 periods

Woodward

Initial profile

x


Outline5

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

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 propagation1

Positive streamer propagation

Charge density (C)

2ns

Zoom

UPWIND

x=0

x=1cm


Positive streamer propagation2

Positive streamer propagation

Charge density (C)

2ns

Zoom

UPWIND

x=0

x=1cm

Charge density (C)

4ns

Quickest

Woodward

Zoom

superbee

minmod


Conclusion

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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