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The geometric theory of charge conservation in particle-in-cell simulations

Part of: Hamiltonian Methods in Plasma Physics

Published online by Cambridge University Press:  27 May 2020

Alexander S. Glasser Open the ORCID record for Alexander S. Glasser [Opens in a new window]  and
Hong Qin
Alexander S. Glasser*
Affiliation:
Princeton Plasma Physics Laboratory, Princeton University, Princeton, NJ 08543, USA Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
Hong Qin
Affiliation:
Princeton Plasma Physics Laboratory, Princeton University, Princeton, NJ 08543, USA Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
*
†Email address for correspondence: [email protected]
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Abstract

In recent years, several gauge-symmetric particle-in-cell (PIC) methods have been developed whose simulations of particles and electromagnetic fields exactly conserve charge. While it is rightly observed that these methods’ gauge symmetry gives rise to their charge conservation, this causal relationship has generally been asserted via ad hoc derivations of the associated conservation laws. In this work, we develop a comprehensive theoretical grounding for charge conservation in gauge-symmetric Lagrangian and Hamiltonian PIC algorithms. For Lagrangian variational PIC methods, we apply Noether’s second theorem to demonstrate that gauge symmetry gives rise to a local charge conservation law as an off-shell identity. For Hamiltonian splitting methods, we show that the momentum map establishes their charge conservation laws. We define a new class of algorithms – gauge-compatible splitting methods – that exactly preserve the momentum map associated with a Hamiltonian system’s gauge symmetry – even after time discretization. This class of algorithms affords splitting schemes a decided advantage over alternative Hamiltonian integrators. We apply this general technique to design a novel, explicit, symplectic, gauge-compatible splitting PIC method, whose momentum map yields an exact local charge conservation law. Our study clarifies the appropriate initial conditions for such schemes and examines their symplectic reduction.

Keywords

plasma simulation
Type
Research Article
Information
Journal of Plasma Physics , Volume 86 , Issue 3 , June 2020 , 835860303
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Abraham, R., Marsden, J. E. & Ratiu, T. S. 1988 Manifolds, Tensor Analysis, and Applications, Second Edition. Springer-Verlag New York, Inc.CrossRefGoogle Scholar
Birdsall, C. K. & Langdon, A. B. 1991 Plasma Physics via Computer Simulation. IOP Publishing Ltd.CrossRefGoogle Scholar
Bossavit, A. 1988 Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism. IEE Proc. A 135 (8), 493–500.Google Scholar
Brading, K. & Brown, H. R.2000 Noether’s theorems and gauge symmetries.arXiv:hep-th/0009058.Google Scholar
Cary, J. R. & Doxas, I. 1993 An explicit symplectic integration scheme for plasma simulations. J. Comput. Phys. 107 (1), 98–104.CrossRefGoogle Scholar
Chandre, C., Guillebon, L. d., Back, A., Tassi, E. & Morrison, P. J. 2013 On the use of projectors for Hamiltonian systems and their relationship with Dirac brackets. J. Phys. A: Math. Theor. 46 (12), 125203.CrossRefGoogle Scholar
Chen, G., Chacón, L. & Barnes, D. C. 2011 An energy- and charge-conserving, implicit, electrostatic particle-in-cell algorithm. J. Comput. Phys. 230 (18), 7018–7036.CrossRefGoogle Scholar
Chen, Y. & Parker, S. E. 2003 A $\unicode[STIX]{x1D6FF}$f particle method for gyrokinetic simulations with kinetic electrons and electromagnetic perturbations. J. Comput. Phys. 189 (2), 463–475.CrossRefGoogle Scholar
Cohen, B. I., Langdon, A. B. & Friedman, A. 1982 Implicit time integration for plasma simulation. J. Comput. Phys. 46 (1), 15–38.CrossRefGoogle Scholar
Cohen, B. I., Langdon, A. B., Hewett, D. W. & Procassini, R. J. 1989 Performance and optimization of direct implicit particle simulation. J. Comput. Phys. 81 (1), 151–168.CrossRefGoogle Scholar
Dawson, J. M. 1983 Particle simulation of plasmas. Rev. Mod. Phys. 55 (2), 403–447.CrossRefGoogle Scholar
Decyk, V. K. 1995 Skeleton PIC codes for parallel computers. Comput. Phys. Commun. 87 (1), 87–94.CrossRefGoogle Scholar
Desbrun, M., Hirani, A. N., Leok, M. & Marsden, J. E.2005 Discrete exterior calculus. arXiv:math/0508341.Google Scholar
Desbrun, M., Kanso, E. & Tong, Y. 2006 Discrete differential forms for computational modeling. In ACM SIGGRAPH 2006 Courses. ACM.Google Scholar
Eastwood, J. W. 1991 The virtual particle electromagnetic particle-mesh method. Comput. Phys. Commun. 64 (2), 252–266.CrossRefGoogle Scholar
Elcott, S. & Schröder, P. 2005 Building your own DEC at home. In ACM SIGGRAPH 2005 Courses. ACM.Google Scholar
Esirkepov, T. 2001 Exact charge conservation scheme for Particle-in-Cell simulation with an arbitrary form-factor. Comput. Phys. Commun. 135 (2), 144–153.CrossRefGoogle Scholar
Friedman, A., Parker, S. E., Ray, S. L. & Birdsall, C. K. 1991 Multi-scale particle-in-cell plasma simulation. J. Comput. Phys. 96 (1), 54–70.CrossRefGoogle Scholar
Grote, D. P., Friedman, A., Haber, I., Fawley, W. & Vay, J. L. 1998 New developments in WARP: progress toward end-to-end simulation. Nucl. Instrum. Meth. Phys. Res. A 415 (1), 428–432.CrossRefGoogle Scholar
Hairer, E., Lubich, C. & Wanner, G. 2006 Geometric numerical integration: structure-preserving algorithms for ordinary differential equations. In Springer Series in Computational Mathematics, vol. 31. Springer.Google Scholar
He, Y., Qin, H., Sun, Y., Xiao, J., Zhang, R. & Liu, J. 2015 Hamiltonian time integrators for Vlasov–Maxwell equations. Phys. Plasmas 22 (12), 124503.CrossRefGoogle Scholar
He, Y., Sun, Y., Qin, H. & Liu, J. 2016 Hamiltonian particle-in-cell methods for Vlasov–Maxwell equations. Phys. Plasmas 23 (9), 092108.CrossRefGoogle Scholar
Hirani, A. N.2003 Discrete exterior calculus. PhD thesis, California Institute of Technology.Google Scholar
Hockney, R. W. & Eastwood, J. W. 1988 Computer Simulation Using Particles. CRC Press.CrossRefGoogle Scholar
Huang, C., Decyk, V. K., Ren, C., Zhou, M., Lu, W., Mori, W. B., Cooley, J. H., Antonsen, T. M. & Katsouleas, T. 2006 QUICKPIC: A highly efficient particle-in-cell code for modeling wakefield acceleration in plasmas. J. Comput. Phys. 217 (2), 658–679.CrossRefGoogle Scholar
Hydon, P. E. & Mansfield, E. L. 2011 Extensions of Noether’s second theorem: from continuous to discrete systems. Proc. R. Soc. Lond. A 467 (2135), 3206–3221.CrossRefGoogle Scholar
Iwinski, Z. & Turski, K. 1976 Canonical theories of systems interacting electromagnetically. Lett. Appl. Engng Sci. 4 (3), 179–191.Google Scholar
Kraus, M., Kormann, K., Morrison, P. J. & Sonnendrücker, E. 2017 GEMPIC: geometric electromagnetic particle-in-cell methods. J. Plasma Phys. 83 (4), 905830401.CrossRefGoogle Scholar
Langdon, A. B., Cohen, B. I. & Friedman, A. 1983 Direct implicit large time-step particle simulation of plasmas. J. Comput. Phys. 51 (1), 107–138.CrossRefGoogle Scholar
Lee, W. W. 1983 Gyrokinetic approach in particle simulation. Phys. Fluids 26 (2), 556–562.CrossRefGoogle Scholar
Liewer, P. C. & Decyk, V. K. 1989 A general concurrent algorithm for plasma particle-in-cell simulation codes. J. Comput. Phys. 85 (2), 302–322.CrossRefGoogle Scholar
Marsden, J. E. & Ratiu, T. 1986 Reduction of Poisson manifolds. Lett. Math. Phys. 11 (2), 161–169.CrossRefGoogle Scholar
Marsden, J. & Ratiu, T. 1999 Introduction to Mechanics and Symmetry, Texts in Applied Mathematics, vol. 17. Springer-Verlag New York, Inc.CrossRefGoogle Scholar
Marsden, J. & Weinstein, A. 1974 Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5 (1), 121–130.CrossRefGoogle Scholar
Marsden, J. E. & Weinstein, A. 1982 The Hamiltonian structure of the Maxwell–Vlasov equations. Physica D 4 (3), 394.Google Scholar
Morrison, P. J. 1980 The Maxwell–Vlasov equations as a continuous Hamiltonian system. Phys. Lett. A 80 (5), 383–386.CrossRefGoogle Scholar
Morrison, P. J. 1982 Poisson brackets for fluids and plasmas. In AIP Conference Proceedings, vol. 88, pp. 13–46. AIP.Google Scholar
Morrison, P. J. 1993 Hamiltonian description of the ideal fluid. In Geometrical Methods in Fluid Dynamics: 1993 Summer Study Program in Geophysical Fluid Dynamics. Woods Hole Oceanographic Institution.Google Scholar
Morrison, P. J. 2013 A general theory for gauge-free lifting. Phys. Plasmas 20 (1), 012104.CrossRefGoogle Scholar
Nieter, C. & Cary, J. R. 2004 VORPAL: a versatile plasma simulation code. J. Comput. Phys. 196 (2), 448–473.CrossRefGoogle Scholar
Okuda, H. 1972 Nonphysical noises and instabilities in plasma simulation due to a spatial grid. J. Comput. Phys. 10, 475–486.CrossRefGoogle Scholar
Olver, P. J. 1986 Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, vol. 107. Springer.CrossRefGoogle Scholar
Parker, S. E., Lee, W. W. & Santoro, R. A. 1993 Gyrokinetic simulation of ion temperature gradient driven turbulence in 3D toroidal geometry. Phys. Rev. Lett. 71 (13), 2042–2045.CrossRefGoogle ScholarPubMed
Pukhov, A. 2016 Particle-in-cell codes for plasma-based particle acceleration. CERN Yellow Reports 1 (0), 181.Google Scholar
Qiang, J., Ryne, R. D., Habib, S. & Decyk, V. 2000 An object-oriented parallel particle-in-cell code for beam dynamics simulation in linear accelerators. J. Comput. Phys. 163 (2), 434–451.CrossRefGoogle Scholar
Qin, H., Davidson, R. C. & Lee, W. W.-l. 2000a Three-dimensional multispecies nonlinear perturbative particle simulations of collective processes in intense particle beams. Phys. Rev. ST Accel. Beams 3 (8), 084401.Google Scholar
Qin, H., Davidson, R. C. & Lee, W. W.-l. 2000b 3D nonlinear perturbative particle simulations of two-stream collective processes in intense particle beams. Phys. Lett. A 272 (5), 389–394.CrossRefGoogle Scholar
Qin, H., Davidson, R. C., Lee, W. W.-l. & Kolesnikov, R. 2001 3D multispecies nonlinear perturbative particle simulations of collective processes in intense particle beams for heavy ion fusion. Nucl. Instrum. Meth. Phys. Res. A 464 (1), 477–483.CrossRefGoogle Scholar
Qin, H., Liu, J., Xiao, J., Zhang, R., He, Y., Wang, Y., Sun, Y., Burby, J. W., Ellison, L. & Zhou, Y. 2016 Canonical symplectic particle-in-cell method for long-term large-scale simulations of the Vlasov–Maxwell equations. Nucl. Fusion 56 (1), 014001.CrossRefGoogle Scholar
Souriau, J.-M. 1970 Structure des systmes dynamiques. Dunod.Google Scholar
Squire, J., Qin, H. & Tang, W. M. 2012 Geometric integration of the Vlasov–Maxwell system with a variational particle-in-cell scheme. Phys. Plasmas 19 (8), 084501.CrossRefGoogle Scholar
Stern, A., Tong, Y., Desbrun, M. & Marsden, J. E. 2015 Geometric computational electrodynamics with variational integrators and discrete differential forms. In Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden (ed. Chang, D. E., Holm, D. D., Patrick, G. & Ratiu, T.), pp. 437–475. Springer New York.CrossRefGoogle Scholar
Vay, J.-L., Colella, P., McCorquodale, P., Straalen, B. V., Friedman, A. & Grote, D. P. 2002 Mesh refinement for particle-in-cell plasma simulations: applications to and benefits for heavy ion fusion. Laser Part. Beams 20 (4), 569–575.CrossRefGoogle Scholar
Villasenor, J. & Buneman, O. 1992 Rigorous charge conservation for local electromagnetic field solvers. Comput. Phys. Commun. 69 (2–3), 306–316.CrossRefGoogle Scholar
Xiao, J. & Qin, H. 2019 Field theory and a structure-preserving geometric particle-in-cell algorithm for drift wave instability and turbulence. Nucl. Fusion 59 (10), 106044.CrossRefGoogle Scholar
Xiao, J., Qin, H. & Liu, J. 2018 Structure-preserving geometric particle-in-cell methods for Vlasov–Maxwell systems. Plasma Sci. Technol. 20 (11), 110501.CrossRefGoogle Scholar
Xiao, J., Qin, H., Liu, J., He, Y., Zhang, R. & Sun, Y. 2015 Explicit high-order non-canonical symplectic particle-in-cell algorithms for Vlasov–Maxwell systems. Phys. Plasmas 22 (11), 112504.CrossRefGoogle Scholar