![Page 1: Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20,](https://reader030.vdocument.in/reader030/viewer/2022032804/56649edd5503460f94bedd1a/html5/thumbnails/1.jpg)
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
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Institut für Aerodynamik und Gasdynamik
Universität Stuttgart
The 30th International Electric Propulsion Conference, Florence, Italy
September 17-20, 2007
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
![Page 3: Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20,](https://reader030.vdocument.in/reader030/viewer/2022032804/56649edd5503460f94bedd1a/html5/thumbnails/3.jpg)
Institut für Aerodynamik und Gasdynamik
Universität Stuttgart
The 30th International Electric Propulsion Conference, Florence, Italy
September 17-20, 2007
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
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Institut für Aerodynamik und Gasdynamik
Universität Stuttgart
The 30th International Electric Propulsion Conference, Florence, Italy
September 17-20, 2007
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
,
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Institut für Aerodynamik und Gasdynamik
Universität Stuttgart
The 30th International Electric Propulsion Conference, Florence, Italy
September 17-20, 2007
Lorentz Equation of Motion
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 :
•
![Page 6: Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20,](https://reader030.vdocument.in/reader030/viewer/2022032804/56649edd5503460f94bedd1a/html5/thumbnails/6.jpg)
Institut für Aerodynamik und Gasdynamik
Universität Stuttgart
The 30th International Electric Propulsion Conference, Florence, Italy
September 17-20, 2007
Lorentz Equation of Motion
20
2
1
0
1
1
01
0 00 2
1ttUttUUtv tt
- Recursive second order scheme for the velocity :
•
2
1
1
0
1
11
1
0000 ttt UUU
002
31
1
0UU
ct
![Page 7: Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20,](https://reader030.vdocument.in/reader030/viewer/2022032804/56649edd5503460f94bedd1a/html5/thumbnails/7.jpg)
Institut für Aerodynamik und Gasdynamik
Universität Stuttgart
The 30th International Electric Propulsion Conference, Florence, Italy
September 17-20, 2007
Lorentz Equation of Motion
20
2
1
0
1
1
01
0 00 2
1ttUttUUtv tt
•
- Recursive second order scheme for the velocity :
•
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
![Page 8: Institut für Aerodynamik und Gasdynamik Universität Stuttgart The 30 th International Electric Propulsion Conference, Florence, Italy September 17-20,](https://reader030.vdocument.in/reader030/viewer/2022032804/56649edd5503460f94bedd1a/html5/thumbnails/8.jpg)
Institut für Aerodynamik und Gasdynamik
Universität Stuttgart
The 30th International Electric Propulsion Conference, Florence, Italy
September 17-20, 2007
Lorentz Equation of Motion
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
•
•
- Recursive second order scheme for the velocity :
2
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Institut für Aerodynamik und Gasdynamik
Universität Stuttgart
The 30th International Electric Propulsion Conference, Florence, Italy
September 17-20, 2007
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
,
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Institut für Aerodynamik und Gasdynamik
Universität Stuttgart
The 30th International Electric Propulsion Conference, Florence, Italy
September 17-20, 2007
- 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
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Institut für Aerodynamik und Gasdynamik
Universität Stuttgart
The 30th International Electric Propulsion Conference, Florence, Italy
September 17-20, 2007
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
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Institut für Aerodynamik und Gasdynamik
Universität Stuttgart
The 30th International Electric Propulsion Conference, Florence, Italy
September 17-20, 2007
- 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
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Institut für Aerodynamik und Gasdynamik
Universität Stuttgart
The 30th International Electric Propulsion Conference, Florence, Italy
September 17-20, 2007
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
Benchmark 2: Relativistic Particle Motion in B-Field
- 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
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Institut für Aerodynamik und Gasdynamik
Universität Stuttgart
The 30th International Electric Propulsion Conference, Florence, Italy
September 17-20, 2007
Nbr of Points
se
ucl
.No
rm
101 102 10310-11
10-9
10-7
10-5
10-3
10-1
101
EOC 2EOC 3EOC 4EOC 5
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
Benchmark 2: Relativistic Particle Motion in B-Field
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
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Institut für Aerodynamik und Gasdynamik
Universität Stuttgart
The 30th International Electric Propulsion Conference, Florence, Italy
September 17-20, 2007
- 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
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Institut für Aerodynamik und Gasdynamik
Universität Stuttgart
The 30th International Electric Propulsion Conference, Florence, Italy
September 17-20, 2007
- Initial velocity in x direction of 0.99c - Positive charge with mass of electron in positive em fields- Negative drift velocity and clockwise rotation
Benchmark 3: Relativistic Particle Motion in ExB-Field
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.
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Institut für Aerodynamik und Gasdynamik
Universität Stuttgart
The 30th International Electric Propulsion Conference, Florence, Italy
September 17-20, 2007
- 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
EOC 2EOC 3EOC 4EOC 5
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
Benchmark 3: Relativistic Particle Motion in ExB-Field
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.
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Institut für Aerodynamik und Gasdynamik
Universität Stuttgart
The 30th International Electric Propulsion Conference, Florence, Italy
September 17-20, 2007
- 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