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NEW ALGORITHMS FOR
CYLINDRICAL PIC CODE
______
APPLICATION TO WAKEFIELD
ACCELERATORS
X. Davoine1, R. Lehe2, A. Lifschitz2, V. Malka2
1CEA,
DAM, DIF, F-91297 Arpajon, France
2LOA, 91761 Palaiseau, France
EAAC Workshop – Elba – June 2013
11 août 2017
CEA | 10 AVRIL 2012 | PAGE 1
Objectives
Accurate simulations are needed, fast simulation are useful
3D PIC simulations are required
2D simulations only provide qualitative understanding of the physics
Numerical schemes have to be improved to reduce the numerical
errors1
Development of reduced PIC codes, such as Calder-Circ1, are
useful to run fast simulations
Outline
Presentation of Calder-Circ1 (cylindrical mesh)
Charge conservation with cylindrical mesh
New Maxwell solver to minimize numerical Cerenkov radiation
Form-factor with cylindrical mesh
1A. Lifschitz
11 août 2017
et
al, JoCP 228, 1803 (2009)
EAAC | June 2013 | PAGE 2
Calder-Circ: a quasi-axisymmetric PIC
code for LWFA
11 août 2017
EAAC | June 2013 | PAGE 3
Calder-Circ: a quasi-axisymmetric PIC
code for LWFA
Cylindrical coordinates are used for the fields
A Fourier decomposition is used in the poloidal direction
The physic of LWFA is quasi-axisymmetric:
Only a few modes are needed, the higher modes can be neglected
Only a few “2D” modes are calculated: the simulation is faster
11 août 2017
A. Lifschitz
et al, JoCP 228, 1803 (2009)
EAAC | June 2013 | PAGE 4
Calder-Circ: similar to a 3D simulation but 50x faster
Laser and plasma parameters
• λ0 = 0.8 μm
• τ0 = 30 fs
• w0 = 9 μm
• a0 = 5
• ne = 0.007 nc
50x faster
11 août 2017
EAAC | June 2013 | PAGE 5
Charge conservation with a cylindrical geometry
11 août 2017
EAAC | June 2013 | PAGE 6
Charge conservation with a cylindrical geometry
2 methods can be used to calculate the current:
The current can be calculated from the particles velocities and then
projected on the grid:
J = qv
The current density can be obtained from the charge conservation
equation1:
1Esirkepov,
11 août 2017JoCP
135 144 (2001)
EAAC | June 2013 | PAGE 7
Charge conservation with a cylindrical geometry
2 methods can be used to calculate the current:
The current can be calculated from the particles velocities and then
projected on the grid:
J = qv
The current density can be obtained from the charge conservation
equation1:
1Esirkepov,
11 août 2017JoCP
135 144 (2001)
EAAC | June 2013 | PAGE 8
Charge conservation with a cylindrical geometry
Inversion of the
operator
Adaptation to cylindrical geometry of the methods proposed by Esirkepov1:
1Esirkepov,
11 août 2017JoCP
135 144 (2001)
EAAC | June 2013 | PAGE 9
Charge conservation error ~ numerical precision
Error on the Poisson equation without the charge conservation scheme for
typical LWFA case:
> 10-3
Error with the charge conservation scheme:
10-16
11 août 2017
Propagation length [mm]
EAAC | June 2013 |
PAGE 10
Improved Maxwell solver
Suppression of the numerical Cerenkov radiation
11 août 2017
EAAC | June 2013 | PAGE 11
Improved Maxwell solver
Suppression of the numerical Cerenkov radiation
The numerical Cerenkov radiation
can degrade the electron beam
properties
New Maxwell solver can reduce
the numerical Cerenkov radiation1
1R.
et
11 Lehe
août 2017
al., PRSTAB 16 021301 (2013)
EAAC | June 2013 | PAGE 12
Improved Maxwell solver
Suppression of the numerical Cerenkov radiation
Development of a new scheme to solve the Maxwell equations (R. Lehe - LOA)
LWFA simulation with the previous
scheme
11 août 2017
LWFA simulation with the new
scheme
EAAC | June 2013 | PAGE 13
Improved Maxwell solver
Suppression of the numerical Cerenkov radiation
Development of a new scheme to solve the Maxwell equations (R. Lehe - LOA)
LWFA simulation with the new
scheme
11 août 2017
EAAC | June 2013 | PAGE 14
Form-factor with cylindrical mesh
11 août 2017
EAAC | June 2013 | PAGE 15
Linear form-factor are not fully satisfying
dx
j+1
dr
SK(x)
j
r
x
1
k
k+1
0
K-1
11
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août 2017
2017
K
K+1
EAAC | June 2013 | PAGE 16
x
Linear form-factor are not fully satisfying
dx
j+1
dr
SK(x)
j
r
x
1
k
k+1
0
J.P. Verboncoeur, JoCP 174, 421 (2001)
K-1
K
K+1
Density profile calculated with the exact volume
Density profile calculated with the corrected volume
11
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août 2017
2017
EAAC | June 2013 | PAGE 17
x
Proposed form-factor to use the exact cell volumes
A quadratic spline should be use in the radial direction
Sj(r)
1
0
j-1
11 août 2017
j
j+1
r
EAAC | June 2013 | PAGE 18
Conclusion
Recent development in Calder-Circ:
New form-factor
Charge conservation algorithm
New Maxwell solver to reduce the numerical Cerenkov radiation
Next developments:
Form-factor at 2nd and 3rd order
Ionization by fields
Interest:
Better modeling of LWFA
Possibility to model plasma with higher density (near critical density
?)
11 août 2017
EAAC | June 2013 | PAGE 19