Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
153 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Instantaneous control strategies for magnetically confined fusion plasma (2403.16254v1)

Published 24 Mar 2024 in math.NA and cs.NA

Abstract: The principle behind magnetic fusion is to confine high temperature plasma inside a device in such a way that the nuclei of deuterium and tritium joining together can release energy. The high temperatures generated needs the plasma to be isolated from the wall of the device to avoid damages and the scope of external magnetic fields is to achieve this goal. In this paper, to face this challenge from a numerical perspective, we propose an instantaneous control mathematical approach to steer a plasma into a given spatial region. From the modeling point of view, we focus on the Vlasov equation in a bounded domain with self induced electric field and an external strong magnetic field. The main feature of the control strategy employed is that it provides a feedback on the equation of motion based on an instantaneous prediction of the discretized system. This permits to directly embed the minimization of a given cost functional into the particle interactions of the corresponding Vlasov model. The numerical results demonstrate the validity of our control approach and the capability of an external magnetic field, even if in a simplified setting, to lead the plasma far from the boundaries.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (63)
  1. Mean field control hierarchy. Applied Mathematics & Optimization, 76:93–135, 2017.
  2. Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems. Applied Mathematics and Computation, 354:460–477, 2019.
  3. A variational sheath model for stationary gyrokinetic Vlasov–Poisson equations. ESAIM: Mathematical Modelling and Numerical Analysis, 55(6):2609–2642, 2021.
  4. Controlling a Vlasov-Poisson plasma by a Particle-in-Cell method based on a Monte Carlo framework. arXiv preprint arXiv:2304.02083, 2023.
  5. An asymptotically stable semi-Lagrangian scheme in the quasi-neutral limit. Journal of Scientific Computing, 41:341–365, 2009.
  6. A model hierarchy for ionospheric plasma modeling. Mathematical Models and Methods in Applied Sciences, 14(03):393–415, 2004.
  7. J. A. Bittencourt. Fundamentals of plasma physics. Springer Science & Business Media, 2013.
  8. J. Blum. Numerical simulation and optimal control in plasma physics. New York, NY; John Wiley and Sons Inc., 1989.
  9. Degenerate anisotropic elliptic problems and magnetized plasma simulations. Communications in Computational Physics, 11(1):147–178, 2012.
  10. Instantaneous control of interacting particle systems in the mean-field limit. Journal of Computational Physics, 405:109181, 2020.
  11. Adjoint DSMC for nonlinear Boltzmann equation constrained optimization. Journal of Computational Physics, 439:110404, 2021.
  12. A hybrid method for accelerated simulation of Coulomb collisions in a plasma. Multiscale Modeling & Simulation, 7(2):865–887, 2008.
  13. Time evolution of a Vlasov-Poisson plasma with magnetic confinement. Kinetic and Related Models, 5(4):729–742, 2012.
  14. Optimization of Particle-in-Cell simulations for Vlasov-Poisson system with strong magnetic field. ESAIM: Proceedings and Surveys, 53:177–190, 2016.
  15. F. F. Chen. Introduction to plasma physics and controlled fusion, volume 1. Springer, 1984.
  16. C.-Z. Cheng and G. Knorr. The integration of the Vlasov equation in configuration space. Journal of Computational Physics, 22(3):330–351, 1976.
  17. Discontinuous Galerkin methods for the Vlasov–Maxwell equations. SIAM Journal on Numerical Analysis, 52(2):1017–1049, 2014.
  18. J. Coughlin and J. Hu. Efficient dynamical low-rank approximation for the Vlasov-Ampère-Fokker-Planck system. J. Comput. Phys., 470:Paper No. 111590, 20, 2022.
  19. Quasi-neutral fluid models for current-carrying plasmas. Journal of Computational Physics, 205(2):408–438, 2005.
  20. Multiscale schemes for the BGK–Vlasov–Poisson system in the quasi-neutral and fluid limits. stability analysis and first order schemes. Multiscale Modeling & Simulation, 14(1):65–95, 2016.
  21. N. Crouseilles and F. Filbet. Numerical approximation of collisional plasmas by high order methods. Journal of Computational Physics, 201(2):546–572, 2004.
  22. Conservative semi-Lagrangian schemes for Vlasov equations. Journal of Computational Physics, 229(6):1927–1953, 2010.
  23. I. Davies. Magnetic control of tokamak plasmas through deep reinforcement learning. Bulletin of the American Physical Society, 2023.
  24. Discontinuous Galerkin methods for the multi-dimensional Vlasov-Poisson problem. Mathematical Models and Methods in Applied Sciences, 22(12):1250042, 2012.
  25. P. Degond. Asymptotic-Preserving schemes for fluid models of plasmas. Société mathématique de France, 2013.
  26. P. Degond and F. Deluzet. Asymptotic-preserving methods and multiscale models for plasma physics. Journal of Computational Physics, 336:429–457, 2017.
  27. Efficient parallelization for 3d-3v sparse grid Particle-in-Cell: Shared memory architectures. Journal of Computational Physics, 480:112022, 2023.
  28. Numerical methods and macroscopic models of magnetically confined low temperature plasmas. Kinetic and Related Models, 16(5):624–653, 2023.
  29. Direct simulation Monte Carlo schemes for Coulomb interactions in plasmas. Communications in Applied and Industrial Mathematics, 1(1):p–72, 2010.
  30. Numerical methods for plasma physics in collisional regimes. Journal of Plasma Physics, 81(1):305810106, 2015.
  31. An asymptotic preserving automatic domain decomposition method for the Vlasov–Poisson–BGK system with applications to plasmas. Journal of Computational Physics, 274:122–139, 2014.
  32. G. Dimarco and L. Pareschi. Numerical methods for kinetic equations. Acta Numer., 23:369–520, 2014.
  33. Higher-order time integration of Coulomb collisions in a plasma using Langevin equations. Journal of Computational Physics, 242:561–580, 2013.
  34. Suppressing instability in a Vlasov–Poisson system by an external electric field through constrained optimization. Journal of Computational Physics, 498:112662, 2024.
  35. Computational challenges in magnetic-confinement fusion physics. Nature Physics, 12(5):411–423, 2016.
  36. F. Filbet and L. M. Rodrigues. Asymptotically stable Particle-in-Cell methods for the Vlasov–Poisson system with a strong external magnetic field. SIAM Journal on Numerical Analysis, 54(2):1120–1146, 2016.
  37. F. Filbet and L. M. Rodrigues. Asymptotically preserving Particle-in-Cell methods for inhomogeneous strongly magnetized plasmas. SIAM Journal on Numerical Analysis, 55(5):2416–2443, 2017.
  38. F. Filbet and E. Sonnendrücker. Numerical methods for the Vlasov equation. In Numerical Mathematics and Advanced Applications: Proceedings of ENUMATH 2001 the 4th European Conference on Numerical Mathematics and Advanced Applications Ischia, July 2001, pages 459–468. 2003.
  39. F. Filbet and T. Xiong. Conservative discontinuous Galerkin/Hermite spectral method for the Vlasov–Poisson system. Communications on Applied Mathematics and Computation, pages 1–26, 2020.
  40. F. Filbet and C. Yang. Numerical simulations to the Vlasov-Poisson system with a strong magnetic field. HAL, 2018.
  41. M. Fornasier and F. Solombrino. Mean-field optimal control. ESAIM: Control, Optimisation and Calculus of Variations, 20(4):1123–1152, 2014.
  42. P. R. Garabedian. Computational mathematics and physics of fusion reactors. Proceedings of the National Academy of Sciences, 100(24):13741–13745, 2003.
  43. An Eulerian code for the study of the drift-kinetic Vlasov equation. Journal of Computational Physics, 108(1):105–121, 1993.
  44. Accuracy and convergence of coupled finite-volume/Monte Carlo codes for plasma edge simulations of nuclear fusion reactors. J. Comput. Phys., 322:162–182, 2016.
  45. V. Grandgirard and Y. Sarazin. Gyrokinetic simulations of magnetic fusion plasmas. pages 91–176. 2013.
  46. Hamiltonian Particle-in-Cell methods for Vlasov–Poisson equations. Journal of Computational Physics, 467:111472, 2022.
  47. W. W. Hager. Runge-Kutta methods in optimal control and the transformed adjoint system. Numerische Mathematik, 87:247–282, 2000.
  48. E. Hairer and C. Lubich. Energy behaviour of the Boris method for charged-particle dynamics. BIT Numerical Mathematics, 58:969–979, 2018.
  49. D. Han-Kwan. On the confinement of a tokamak plasma. SIAM journal on mathematical analysis, 42(6):2337–2367, 2010.
  50. Implicit-explicit Runge–Kutta schemes for numerical discretization of optimal control problems. SIAM Journal on Numerical Analysis, 51(4):1875–1899, 2013.
  51. A hybrid fluid-kinetic model for hydrogenic atoms in the plasma edge of tokamaks based on a micro-macro decomposition of the kinetic equation. J. Comput. Phys., 409:109308, 23, 2020.
  52. A. Jüngel and Y.-J. Peng. A hierarchy of hydrodynamic models for plasmas. quasi-neutral limits in the drift-diffusion equations. Asymptotic Analysis, 28(1):49–73, 2001.
  53. P. Knopf and J. Weber. Optimal control of a Vlasov–Poisson plasma by fixed magnetic field coils. Applied Mathematics & Optimization, 81:961–988, 2020.
  54. T. Laidin and T. Rey. Hybrid kinetic/fluid numerical method for the Vlasov-Poisson-BGK equation in the diffusive scaling. In Finite volumes for complex applications X—volume 2, hyperbolic and related problems, volume 433 of Springer Proc. Math. Stat., pages 229–237. Springer, Cham, [2023] ©2023.
  55. Stochastic Galerkin particle methods for kinetic equations of plasmas with uncertainties. Journal of Computational Physics, 479:112011, 2023.
  56. S. Ravindran. Real-time computational algorithm for optimal control of an MHD flow system. SIAM Journal on Scientific Computing, 26(4):1369–1388, 2005.
  57. Multilevel Monte Carlo simulation of Coulomb collisions. Journal of Computational Physics, 274:140–157, 2014.
  58. G. Russo and F. Filbet. Semi-Lagrangian schemes applied to moving boundary problems for the BGK model of rarefied gas dynamics. Kinetic and related models, 2(1):231–250, 2009.
  59. J. M. Sanz-Serna. Symplectic Runge–Kutta schemes for adjoint equations, automatic differentiation, optimal control, and more. SIAM review, 58(1):3–33, 2016.
  60. The semi-Lagrangian method for the numerical resolution of the Vlasov equation. Journal of computational physics, 149(2):201–220, 1999.
  61. J. Weber. Optimal control of the two-dimensional Vlasov-Maxwell system. ESAIM: Control, Optimisation and Calculus of Variations, 27:S19, 2021.
  62. T. Xiao and M. Frank. A stochastic kinetic scheme for multi-scale plasma transport with uncertainty quantification. Journal of Computational Physics, 432:110139, 2021.
  63. C. Yang and F. Filbet. Conservative and non-conservative methods based on Hermite weighted essentially non-oscillatory reconstruction for Vlasov equations. Journal of Computational Physics, 279:18–36, 2014.
Citations (2)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now