a high order relativistic particle push method for pic simulations

Institut für Aerodynamik und Gasdynamik

Universität Stuttgart

The 30th International Electric Propulsion Conference, Florence, Italy

September 17-20, 2007

A High Order Relativistic Particle Push Method for PIC Simulations

M.Quandt, C.-D. Munz, R.Schneider





Overview

Motivation Mathematical Model Results of Convergence Studies

Non Relativistic Motion in Time varying E-Field Relativistic B-Field Motion Relativistic ExB Drift

Conclusions and future Works





Motivation

- Coupled PIC/DSMC/FP PicLas-Code concept to study electric propulsion systems- High Order PIC method for the self-consistent solution of the Maxwell-Vlasov equations- For Consistency: High Order Lorentz solver which increase accuracy and efficiency





Lorentz Equation of Motion

txBtvtxEqtvtmt

,,0

txtUttxtU ,, 1

tUttv 1

- Truncated Taylor series expansion :

0

0

10!

1

tt

p

Ut

tttv

0

0

110!1

1

tt

p

Ut

ttts

21

c

UU

- Mathematical model :

Bm

qE

m

q

00

,






1

1

0

1

11

1

0000 ttt UUU

2 20

2

1

0

1

1

01

0 00 2

1ttUttUUtv tt

- Recursive second order scheme for the velocity :

•






20

2

1

0

1

1

01

0 00 2

1ttUttUUtv tt


•

2

1

1

0

1

11

1

0000 ttt UUU

002

31

1

0UU

ct






20

2

1

0

1

1

01

0 00 2

1ttUttUUtv tt

•


•

2

1

1

0

1

11

1

0000 ttt UUU

002

31

1

0UU

ct

2

11

1

1

0

2

12

1

0000002 ttttt UUUU






1

1

0

1

11

1

0000 ttt UUU

002

31

1

0UU

ct

2

11

1

1

0

2

12

1

0000002 ttttt UUUU

1

2

30

2

4

52

1

0

0

0

3t

tt UU

cUU

c

1

1

0

0

1

11

2

00000 ttttt UUU

20

2

1

0

1

1

01

0 00 2

1ttUttUUtv tt

•

•


2





Convergence Studies

Set up of numerical experiments :

- fixed final time

- at compute the norm

- experimental order of convergence

fT pP

f nn

Tt - number of discretization points

fT 2

fexactfnump TQTQEN

p

P

nn

ENEN

EOC0

0

log

log

vsQ

,





- Lorentz factor

- Variation in amplitude, phase shift and angular rate

- All derivatives can be computed immediately

- Parameters :

Benchmark 1: Non-Relativistic Particle Motion

zzz

yyy

xxx

t

t

t

t

sin

sin

sin

ttvt

, 2/3 ,3/2 ,2 zyx, 1 zyx 2/1 zyx

0

t

1)( t





Benchmark 1: Non-Relativistic Particle Motion

The result of EOC are in good agreement with expected formel order.

Points [-]

ve

ucl

.No

rm[-

]

101 102 10310-13

10-11

10-9

10-7

10-5

10-3

10-1

EOC 2EOC 3EOC 4EOC 5

-1.05

-1.025

-1

-0.975

-0.95

z

-1.02-1

-0.98

x-1.2

-1

-0.8

y

XY

Z

The 3D Lissajous trajectories; Line: exact; dots: numerical





- Lorentz factor gamma > 1 and constant in time

- Particle trajectory in xy-plane with constant B-Field

- Parameters of a positive charge and mass of electron

Benchmark 2: Relativistic Particle Motion in B-Field

zgyro 1

0

t Tzt ,0,0

ttvtv

1

,

000

0cossin

0sincos

0

0

0

z

y

x

gyrogyro

gyrogyro

v

v

v

tt

tt

tU

)( 0t





x [m]

y[m

]

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

-0.025

-0.020

-0.015

-0.010

-0.005

0.000


- Initial Values of positive charge and mass of electron, positive B-field and velocity

- Particle trajectory calculated with a third and fifth order scheme for 10 cycles

- Particle trajectory deviates due to the error accumulation in time

Particle trajectory with 160 intervals starts at (0,0). Line: exact; filled circles:third order scheme





Nbr of Points

se

ucl

.No

rm

101 102 10310-11

10-9

10-7

10-5

10-3

10-1

101


x [m]

y[m

]

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

-0.025

-0.020

-0.015

-0.010

-0.005

0.000

x [m]

y[m

]

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

-0.025

-0.020

-0.015

-0.010

-0.005

0.000


Slope of the graphs correspond to the experimental order of convergence.

Improved Solution: fifth order

- Initial Values of positive charge and mass of electron, positive B-field and velocity

- Particle trajectory calculated with a third and fifth order scheme for 10 cycles

- Particle trajectory deviates due to the error accumulation in time

Particle trajectory with 160 intervals starts at (0,0). Line: exact; filled circles:third order scheme





- Problem reduction through Lorentz transformation for

- Same equation as previously but now in primed reference system

- Back transformation yields to ExB motion

Benchmark 3: Relativistic Particle Motion in ExB-Field

Lorentz transformation with velocity into primed reference system.

txBtxE , ; ,

BcE

0 ; ; 0 || BBBE





- Initial velocity in x direction of 0.99c - Positive charge with mass of electron in positive em fields- Negative drift velocity and clockwise rotation


x [m]

y[m

]

-0.2 -0.1 0 0.1 0.2

-3.00

-2.00

-1.00

0.00

Third order approximation: visible deviation from the analytic solution.





- Initial velocity in x direction of 0.99c - Positive charge with mass of electron in positive em fields- Negative drift velocity and clockwise rotation

Nbr of Points

se

ucl

.No

rm

102 10310-11

10-9

10-7

10-5

10-3

10-1


x [m]

y[m

]

-0.2 -0.1 0 0.1 0.2

-3.00

-2.00

-1.00

0.00

x [m]

y[m

]

-0.2 -0.1 0 0.1 0.2

-3.0

-2.0

-1.0

0.0


Expected orders of convergence reached with sufficent number points.

Third order approximation: visible deviation from the analytic solution.

Fifth order scheme: Perfect agreement between exact and numerical solution.





- Taylor series expansion in time applied to the Lorentz equation up to order 5

- Tested on 3 benchmark problems and EOC reached finally the expected order

- Extension to higher order schemes (greater equal 6 )

- Stability analysis

- Further benchmark tests- vs usual second order PIC scheme- vs defined problems of the HOUPIC project for accelerators

- Coupling Maxell-Vlasov solver with FP and DSMC modules

- Application to technical devices

(pulsed plasma thruster, high-power high-frequency microwave generation)

Conclusion and Future Works

a high order relativistic particle push method for pic simulations

Documents