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Kinetic Simulation of Collisional Magnetized Plasmas with Semi-implicit Time Integration

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Abstract

Plasmas with varying collisionalities occur in many applications, such as tokamak edge regions, where the flows are characterized by significant variations in density and temperature. While a kinetic model is necessary for weakly-collisional high-temperature plasmas, high collisionality in colder regions render the equations numerically stiff due to disparate time scales. In this paper, we propose an implicit–explicit algorithm for such cases, where the collisional term is integrated implicitly in time, while the advective term is integrated explicitly in time, thus allowing time step sizes that are comparable to the advective time scales. This partitioning results in a more efficient algorithm than those using explicit time integrators, where the time step sizes are constrained by the stiff collisional time scales. We implement semi-implicit additive Runge–Kutta methods in COGENT, a high-order finite-volume gyrokinetic code and test the accuracy, convergence, and computational cost of these semi-implicit methods for test cases with highly-collisional plasmas.

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References

  1. Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit–explicit Runge–Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25(2–3), 151–167 (1997). https://doi.org/10.1016/S0168-9274(97)00056-1

    Article  MathSciNet  MATH  Google Scholar 

  2. Banks, J.W., Brunner, S., Berger, R.L., Tran, T.M.: Vlasov simulations of electron-ion collision effects on damping of electron plasma waves. Phys. Plasmas 23(3), 032108 (2016). https://doi.org/10.1063/1.4943194

    Article  Google Scholar 

  3. Belli, E.A., Candy, J.: Full linearized Fokker–Planck collisions in neoclassical transport simulations. Plasma Phys. Control. Fusion 54(1), 015015 (2012). https://doi.org/10.1088/0741-3335/54/1/015015

    Article  Google Scholar 

  4. Berezin, Y., Khudick, V., Pekker, M.: Conservative finite-difference schemes for the Fokker–Planck equation not violating the law of an increasing entropy. J. Comput. Phys. 69(1), 163–174 (1987). https://doi.org/10.1016/0021-9991(87)90160-4

    Article  MathSciNet  MATH  Google Scholar 

  5. Braginskii, S.I.: Transport processes in a plasma. Rev. Plasma Phys. 1, 205 (1965)

    Google Scholar 

  6. Brunner, S., Tran, T., Hittinger, J.: Numerical implementation of the non-linear Landau collision operator for Eulerian Vlasov simulations. part I: computation of the Rosenbluth potentials. Tech. Rep. LLNL-SR-459135, Lawrence Livermore National Laboratory, Livermore, CA (2010)

  7. Buet, C., Cordier, S.: Conservative and entropy decaying numerical scheme for the isotropic Fokker–Planck–Landau equation. J. Comput. Phys. 145(1), 228–245 (1998). https://doi.org/10.1006/jcph.1998.6015

    Article  MathSciNet  MATH  Google Scholar 

  8. Buet, C., Cordier, S., Degond, P., Lemou, M.: Fast algorithms for numerical, conservative, and entropy approximations of the Fokker–Planck–Landau equation. J. Comput. Phys. 133(2), 310–322 (1997). https://doi.org/10.1006/jcph.1997.5669

    Article  MathSciNet  MATH  Google Scholar 

  9. Butcher, J.: Numerical Methods for Ordinary Differential Equations. Wiley, New York (2003)

    Book  MATH  Google Scholar 

  10. Candy, J., Waltz, R.E.: Anomalous transport scaling in the DIII-D tokamak matched by supercomputer simulation. Phys. Rev. Lett. 91, 045001 (2003). https://doi.org/10.1103/PhysRevLett.91.045001

    Article  Google Scholar 

  11. Casanova, M., Larroche, O., Matte, J.P.: Kinetic simulation of a collisional shock wave in a plasma. Phys. Rev. Lett. 67, 2143–2146 (1991). https://doi.org/10.1103/PhysRevLett.67.2143

    Article  Google Scholar 

  12. Chacón, L., Barnes, D.C., Knoll, D.A., Miley, G.H.: An implicit energy-conservative 2D Fokker–Planck algorithm: I. Difference scheme. J. Comput. Phys. 157(2), 618–653 (2000). https://doi.org/10.1006/jcph.1999.6394

    Article  MathSciNet  MATH  Google Scholar 

  13. Chacón, L., Barnes, D.C., Knoll, D.A., Miley, G.H.: An implicit energy-conservative 2D Fokker–Planck algorithm: II. Jacobian-free Newton–Krylov solver. J. Comput. Phys. 157(2), 654–682 (2000). https://doi.org/10.1006/jcph.1999.6395

    Article  MathSciNet  MATH  Google Scholar 

  14. Chang, C.S., Ku, S.: Spontaneous rotation sources in a quiescent tokamak edge plasma. Phys. Plasmas 15(6), 062510 (2008). https://doi.org/10.1063/1.2937116

    Article  Google Scholar 

  15. Chang, J., Cooper, G.: A practical difference scheme for Fokker–Planck equations. J. Comput. Phys. 6(1), 1–16 (1970). https://doi.org/10.1016/0021-9991(70)90001-X

    Article  MATH  Google Scholar 

  16. Chen, G., Chacón, L.: An energy- and charge-conserving, nonlinearly implicit, electromagnetic 1D-3V Vlasov–Darwin particle-in-cell algorithm. Comput. Phys. Commun. 185(10), 2391–2402 (2014). https://doi.org/10.1016/j.cpc.2014.05.010

    Article  MATH  Google Scholar 

  17. Chen, G., Chacón, L.: A multi-dimensional, energy- and charge-conserving, nonlinearly implicit, electromagnetic Vlasov–Darwin particle-in-cell algorithm. Comput. Phys. Commun. 197, 73–87 (2015). https://doi.org/10.1016/j.cpc.2015.08.008

    Article  MathSciNet  MATH  Google Scholar 

  18. Chen, G., Chacón, L., Barnes, D.: An energy- and charge-conserving, implicit, electrostatic particle-in-cell algorithm. J. Comput. Phys. 230(18), 7018–7036 (2011). https://doi.org/10.1016/j.jcp.2011.05.031

    Article  MathSciNet  MATH  Google Scholar 

  19. Cheng, C., Knorr, G.: The integration of the Vlasov equation in configuration space. J. Comput. Phys. 22(3), 330–351 (1976). https://doi.org/10.1016/0021-9991(76)90053-X

    Article  Google Scholar 

  20. Cohen, R.H., Xu, X.Q.: Progress in kinetic simulation of edge plasmas. Contrib. Plasma Phys. 48(1–3), 212–223 (2008). https://doi.org/10.1002/ctpp.200810038

    Article  Google Scholar 

  21. Colella, P., Dorr, M., Hittinger, J., Martin, D.: High-order, finite-volume methods in mapped coordinates. J. Comput. Phys. 230(8), 2952–2976 (2011). https://doi.org/10.1016/j.jcp.2010.12.044

    Article  MathSciNet  MATH  Google Scholar 

  22. Crouseilles, N., Respaud, T., Sonnendrücker, E.: A forward semi-Lagrangian method for the numerical solution of the Vlasov equation. Comput. Phys. Commun. 180(10), 1730–1745 (2009). https://doi.org/10.1016/j.cpc.2009.04.024

    Article  MathSciNet  MATH  Google Scholar 

  23. Dennis, J., Schnabel, R.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Society for Industrial and Applied Mathematics, Philadelphia (1996). https://doi.org/10.1137/1.9781611971200

    Book  MATH  Google Scholar 

  24. Dorf, M.A., Cohen, R.H., Compton, J.C., Dorr, M., Rognlien, T.D., Angus, J., Krasheninnikov, S., Colella, P., Martin, D., McCorquodale, P.: Progress with the COGENT edge kinetic code: collision operator options. Contrib. Plasma Phys. 52(5–6), 518–522 (2012). https://doi.org/10.1002/ctpp.201210042

    Article  Google Scholar 

  25. Dorf, M.A., Cohen, R.H., Dorr, M., Hittinger, J., Rognlien, T.D.: Progress with the COGENT edge kinetic code: implementing the Fokker–Planck collision operator. Contrib. Plasma Phys. 54(4–6), 517–523 (2014). https://doi.org/10.1002/ctpp.201410023

    Article  Google Scholar 

  26. Dorf, M.A., Cohen, R.H., Dorr, M., Rognlien, T., Hittinger, J., Compton, J., Colella, P., Martin, D., McCorquodale, P.: Simulation of neoclassical transport with the continuum gyrokinetic code COGENT. Phys. Plasmas 20(1), 012513 (2013). https://doi.org/10.1063/1.4776712

    Article  Google Scholar 

  27. Dorr, M.R., Colella, P., Dorf, M.A., Ghosh, D., Hittinger, J.A.F., Schwartz, P.O.: High-order discretization of a gyrokinetic Vlasov model in edge plasma geometry. Submitted (2017). arXiv:1712.01978

  28. Durran, D.R., Blossey, P.N.: Implicit–explicit multistep methods for fast-wave-slow-wave problems. Mon. Weather Rev. 140(4), 1307–1325 (2012). https://doi.org/10.1175/MWR-D-11-00088.1

    Article  Google Scholar 

  29. Epperlein, E.: Implicit and conservative difference scheme for the Fokker–Planck equation. J. Comput. Phys. 112(2), 291–297 (1994). https://doi.org/10.1006/jcph.1994.1101

    Article  MATH  Google Scholar 

  30. Epperlein, E.M., Rickard, G.J., Bell, A.R.: Two-dimensional nonlocal electron transport in laser-produced plasmas. Phys. Rev. Lett. 61, 2453–2456 (1988). https://doi.org/10.1103/PhysRevLett.61.2453

    Article  Google Scholar 

  31. Falgout, R.D., Yang, U.M.: hypre: A Library of High Performance Preconditioners, pp. 632–641. Springer, Berlin (2002). https://doi.org/10.1007/3-540-47789-6_66

    Book  MATH  Google Scholar 

  32. Filbet, F., Pareschi, L.: A numerical method for the accurate solution of the Fokker–Planck–Landau equation in the nonhomogeneous case. J. Comput. Phys. 179(1), 1–26 (2002). https://doi.org/10.1006/jcph.2002.7010

    Article  MathSciNet  MATH  Google Scholar 

  33. Filbet, F., Sonnendrücker, E., Bertrand, P.: Conservative numerical schemes for the Vlasov equation. J. Comput. Phys. 172(1), 166–187 (2001). https://doi.org/10.1006/jcph.2001.6818

    Article  MathSciNet  MATH  Google Scholar 

  34. Ghosh, D., Constantinescu, E.M.: Semi-implicit time integration of atmospheric flows with characteristic-based flux partitioning. SIAM J. Sci. Comput. 38(3), A1848–A1875 (2016). https://doi.org/10.1137/15M1044369

    Article  MathSciNet  MATH  Google Scholar 

  35. Giraldo, F.X., Kelly, J.F., Constantinescu, E.: Implicit–explicit formulations of a three-dimensional nonhydrostatic unified model of the atmosphere (NUMA). SIAM J. Sci. Comput. 35(5), B1162–B1194 (2013). https://doi.org/10.1137/120876034

    Article  MathSciNet  MATH  Google Scholar 

  36. Giraldo, F.X., Restelli, M., Läuter, M.: Semi-implicit formulations of the Navier–Stokes equations: application to nonhydrostatic atmospheric modeling. SIAM J. Sci. Comput. 32(6), 3394–3425 (2010). https://doi.org/10.1137/090775889

    Article  MathSciNet  MATH  Google Scholar 

  37. Grandgirard, V., Brunetti, M., Bertrand, P., Besse, N., Garbet, X., Ghendrih, P., Manfredi, G., Sarazin, Y., Sauter, O., Sonnendrücker, E., Vaclavik, J., Villard, L.: A drift-kinetic semi-Lagrangian 4D code for ion turbulence simulation. J. Comput. Phys. 217(2), 395–423 (2006). https://doi.org/10.1016/j.jcp.2006.01.023

    Article  MathSciNet  MATH  Google Scholar 

  38. Hahm, T.S.: Nonlinear gyrokinetic equations for turbulence in core transport barriers. Phys. Plasmas 3(12), 4658–4664 (1996). https://doi.org/10.1063/1.872034

    Article  Google Scholar 

  39. Heikkinen, J., Kiviniemi, T., Kurki-Suonio, T., Peeters, A., Sipilä, S.: Particle simulation of the neoclassical plasmas. J. Comput. Phys. 173(2), 527–548 (2001). https://doi.org/10.1006/jcph.2001.6891

    Article  MATH  Google Scholar 

  40. Heikkinen, J.A., Henriksson, S., Janhunen, S., Kiviniemi, T.P., Ogando, F.: Gyrokinetic simulation of particle and heat transport in the presence of wide orbits and strong profile variations in the edge plasma. Contrib. Plasma Phys. 46(7–9), 490–495 (2006). https://doi.org/10.1002/ctpp.200610035

    Article  Google Scholar 

  41. Huba, J.D.: NRL plasma formulary. Tech. rep, Naval Research Laboratory, Washington, DC (2016)

  42. Idomura, Y., Ida, M., Kano, T., Aiba, N., Tokuda, S.: Conservative global gyrokinetic toroidal full-f five-dimensional Vlasov simulation. Comput. Phys. Commun. 179(6), 391–403 (2008). https://doi.org/10.1016/j.cpc.2008.04.005

    Article  MathSciNet  MATH  Google Scholar 

  43. Idomura, Y., Ida, M., Tokuda, S.: Conservative gyrokinetic Vlasov simulation. Commun. Nonlinear Sci. Numer. Simul. 13(1), 227–233 (2008). https://doi.org/10.1016/j.cnsns.2007.05.015. (Vlasovia 2006: The Second International Workshop on the Theory and Applications of the Vlasov Equation)

    Article  MATH  Google Scholar 

  44. Jackson, J.D.: Classical Electrodynamics, 3rd edn. Wiley, New York (1999)

    MATH  Google Scholar 

  45. Jacobs, G., Hesthaven, J.: Implicit-explicit time integration of a high-order particle-in-cell method with hyperbolic divergence cleaning. Comput. Phys. Commun. 180(10), 1760–1767 (2009). https://doi.org/10.1016/j.cpc.2009.05.020

    Article  MathSciNet  MATH  Google Scholar 

  46. James, R.: The solution of Poisson’s equation for isolated source distributions. J. Comput. Phys. 25(2), 71–93 (1977). https://doi.org/10.1016/0021-9991(77)90013-4

    Article  MathSciNet  MATH  Google Scholar 

  47. Kadioglu, S.Y., Knoll, D.A., Lowrie, R.B., Rauenzahn, R.M.: A second order self-consistent IMEX method for radiation hydrodynamics. J. Comput. Phys. 229(22), 8313–8332 (2010). https://doi.org/10.1016/j.jcp.2010.07.019

    Article  MathSciNet  MATH  Google Scholar 

  48. Kennedy, C.A., Carpenter, M.H.: Additive Runge–Kutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math. 44(1–2), 139–181 (2003). https://doi.org/10.1016/S0168-9274(02)00138-1

    Article  MathSciNet  MATH  Google Scholar 

  49. Kho, T.H.: Relaxation of a system of charged particles. Phys. Rev. A 32, 666–669 (1985). https://doi.org/10.1103/PhysRevA.32.666

    Article  Google Scholar 

  50. Kingham, R., Bell, A.: An implicit Vlasov–Fokker–Planck code to model non-local electron transport in 2-D with magnetic fields. J. Comput. Phys. 194(1), 1–34 (2004). https://doi.org/10.1016/j.jcp.2003.08.017

    Article  MathSciNet  MATH  Google Scholar 

  51. Knoll, D., Keyes, D.: Jacobian-free Newton–Krylov methods: a survey of approaches and applications. J. Comput. Phys. 193(2), 357–397 (2004). https://doi.org/10.1016/j.jcp.2003.08.010

    Article  MathSciNet  MATH  Google Scholar 

  52. Kumar, H., Mishra, S.: Entropy stable numerical schemes for two-fluid plasma equations. J. Sci. Comput. 52(2), 401–425 (2012). https://doi.org/10.1007/s10915-011-9554-7

    Article  MathSciNet  MATH  Google Scholar 

  53. Landau, L.D.: The kinetic equation in the case of Coulomb interaction. Zh. Eksper. i Teoret. Fiz. 7(2), 203–209 (1937)

    MATH  Google Scholar 

  54. Landreman, M., Ernst, D.R.: New velocity-space discretization for continuum kinetic calculations and Fokker–Planck collisions. J. Comput. Phys. 243, 130–150 (2013). https://doi.org/10.1016/j.jcp.2013.02.041

    Article  MathSciNet  MATH  Google Scholar 

  55. Larroche, O.: Kinetic simulation of a plasma collision experiment. Phys. Fluids B 5(8), 2816–2840 (1993). https://doi.org/10.1063/1.860670

    Article  Google Scholar 

  56. Larroche, O.: Kinetic simulations of fuel ion transport in ICF target implosions. Eur. Phys. J. D At. Mol. Opt. Plasma Phys. 27(2), 131–146 (2003). https://doi.org/10.1140/epjd/e2003-00251-1

    Article  Google Scholar 

  57. Larsen, E., Levermore, C., Pomraning, G., Sanderson, J.: Discretization methods for one-dimensional Fokker–Planck operators. J. Comput. Phys. 61(3), 359–390 (1985). https://doi.org/10.1016/0021-9991(85)90070-1

    Article  MathSciNet  MATH  Google Scholar 

  58. Lee, W.W.: Gyrokinetic approach in particle simulation. Phys. Fluids 26(2), 556–562 (1983). https://doi.org/10.1063/1.864140

    Article  MATH  Google Scholar 

  59. Lemou, M.: Multipole expansions for the Fokker–Planck–Landau operator. Numer. Math. 78(4), 597–618 (1998). https://doi.org/10.1007/s002110050327

    Article  MathSciNet  MATH  Google Scholar 

  60. Lemou, M., Mieussens, L.: Fast implicit schemes for the Fokker–Planck–Landau equation. C. R. Math. 338(10), 809–814 (2004). https://doi.org/10.1016/j.crma.2004.03.010

    Article  MathSciNet  MATH  Google Scholar 

  61. Lemou, M., Mieussens, L.: Implicit schemes for the Fokker–Planck–Landau equation. SIAM J. Sci. Comput. 27(3), 809–830 (2005). https://doi.org/10.1137/040609422

    Article  MathSciNet  MATH  Google Scholar 

  62. Maeyama, S., Ishizawa, A., Watanabe, T.H., Nakajima, N., Tsuji-Iio, S., Tsutsui, H.: A hybrid method of semi-Lagrangian and additive semi-implicit Runge–Kutta schemes for gyrokinetic Vlasov simulations. Comput. Phys. Commun. 183(9), 1986–1992 (2012). https://doi.org/10.1016/j.cpc.2012.04.028

    Article  Google Scholar 

  63. McCorquodale, P., Dorr, M., Hittinger, J., Colella, P.: High-order finite-volume methods for hyperbolic conservation laws on mapped multiblock grids. J. Comput. Phys. 288, 181–195 (2015). https://doi.org/10.1016/j.jcp.2015.01.006

    Article  MathSciNet  MATH  Google Scholar 

  64. McCoy, M., Mirin, A., Killeen, J.: FPPAC: a two-dimensional multispecies nonlinear Fokker–Planck package. Comput. Phys. Commun. 24(1), 37–61 (1981). https://doi.org/10.1016/0010-4655(81)90105-3

    Article  Google Scholar 

  65. Mousseau, V., Knoll, D.: Fully implicit kinetic solution of collisional plasmas. J. Comput. Phys. 136(2), 308–323 (1997). https://doi.org/10.1006/jcph.1997.5736

    Article  MATH  Google Scholar 

  66. Pareschi, L., Russo, G.: Implicit–explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25(1–2), 129–155 (2005). https://doi.org/10.1007/BF02728986

    Article  MathSciNet  MATH  Google Scholar 

  67. Pataki, A., Greengard, L.: Fast elliptic solvers in cylindrical coordinates and the Coulomb collision operator. J. Comput. Phys. 230(21), 7840–7852 (2011). https://doi.org/10.1016/j.jcp.2011.07.005

    Article  MathSciNet  MATH  Google Scholar 

  68. Pernice, M., Walker, H.F.: NITSOL: a Newton iterative solver for nonlinear systems. SIAM J. Sci. Comput. 19(1), 302–318 (1998). https://doi.org/10.1137/S1064827596303843

    Article  MathSciNet  MATH  Google Scholar 

  69. Porter, G.D., Isler, R., Boedo, J., Rognlien, T.D.: Detailed comparison of simulated and measured plasma profiles in the scrape-off layer and edge plasma of DIII-D. Phys. Plasmas 7(9), 3663–3680 (2000). https://doi.org/10.1063/1.1286509

    Article  Google Scholar 

  70. Qiu, J.M., Christlieb, A.: A conservative high order semi-Lagrangian WENO method for the Vlasov equation. J. Comput. Phys. 229(4), 1130–1149 (2010). https://doi.org/10.1016/j.jcp.2009.10.016

    Article  MathSciNet  MATH  Google Scholar 

  71. Qiu, J.M., Shu, C.W.: Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: theoretical analysis and application to the Vlasov–Poisson system. J. Comput. Phys. 230(23), 8386–8409 (2011). https://doi.org/10.1016/j.jcp.2011.07.018

    Article  MathSciNet  MATH  Google Scholar 

  72. Rosenbluth, M.N., MacDonald, W.M., Judd, D.L.: Fokker–Planck equation for an inverse-square force. Phys. Rev. 107, 1–6 (1957). https://doi.org/10.1103/PhysRev.107.1

    Article  MathSciNet  MATH  Google Scholar 

  73. Rossmanith, J.A., Seal, D.C.: A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations. J. Comput. Phys. 230(16), 6203–6232 (2011). https://doi.org/10.1016/j.jcp.2011.04.018

    Article  MathSciNet  MATH  Google Scholar 

  74. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2003)

    Book  MATH  Google Scholar 

  75. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986). https://doi.org/10.1137/0907058

    Article  MathSciNet  MATH  Google Scholar 

  76. Salmi, S., Toivanen, J., von Sydow, L.: An IMEX-scheme for pricing options under stochastic volatility models with jumps. SIAM J. Sci. Comput. 36(5), B817–B834 (2014). https://doi.org/10.1137/130924905

    Article  MathSciNet  MATH  Google Scholar 

  77. Scott, B.: Gyrokinetic study of the edge shear layer. Plasma Phys. Controll. Fusion 48(5A), A387 (2006)

    Article  Google Scholar 

  78. Taitano, W., Chacón, L., Simakov, A.: An adaptive, conservative 0D-2V multispecies Rosenbluth–Fokker–Planck solver for arbitrarily disparate mass and temperature regimes. J. Comput. Phys. 318, 391–420 (2016). https://doi.org/10.1016/j.jcp.2016.03.071

    Article  MathSciNet  MATH  Google Scholar 

  79. Taitano, W., Chacón, L., Simakov, A., Molvig, K.: A mass, momentum, and energy conserving, fully implicit, scalable algorithm for the multi-dimensional, multi-species Rosenbluth–Fokker–Planck equation. J. Comput. Phys. 297, 357–380 (2015). https://doi.org/10.1016/j.jcp.2015.05.025

    Article  MathSciNet  MATH  Google Scholar 

  80. Thomas, A., Tzoufras, M., Robinson, A., Kingham, R., Ridgers, C., Sherlock, M., Bell, A.: A review of Vlasov–Fokker–Planck numerical modeling of inertial confinement fusion plasma. J. Comput. Phys. 231(3), 1051–1079 (2012). https://doi.org/10.1016/j.jcp.2011.09.028. (Special issue: computational plasma physics)

    Article  MathSciNet  MATH  Google Scholar 

  81. Thomas, A.G.R., Kingham, R.J., Ridgers, C.P.: Rapid self-magnetization of laser speckles in plasmas by nonlinear anisotropic instability. New J. Phys. 11(3), 033001 (2009)

    Article  Google Scholar 

  82. Wolke, R., Knoth, O.: Implicit–explicit Runge–Kutta methods applied to atmospheric chemistry-transport modelling. Environ. Model. Softw. 15(6–7), 711–719 (2000). https://doi.org/10.1016/S1364-8152(00)00034-7

    Article  Google Scholar 

  83. Xu, X., Bodi, K., Cohen, R., Krasheninnikov, S., Rognlien, T.: TEMPEST simulations of the plasma transport in a single-null tokamak geometry. Nucl. Fusion 50(6), 064003 (2010)

    Article  Google Scholar 

  84. Xu, X., Xiong, Z., Dorr, M., Hittinger, J., Bodi, K., Candy, J., Cohen, B., Cohen, R., Colella, P., Kerbel, G., Krasheninnikov, S., Nevins, W., Qin, H., Rognlien, T., Snyder, P., Umansky, M.: Edge gyrokinetic theory and continuum simulations. Nucl. Fusion 47(8), 809 (2007)

    Article  Google Scholar 

  85. Xu, X.Q., Xiong, Z., Gao, Z., Nevins, W.M., McKee, G.R.: TEMPEST simulations of collisionless damping of the geodesic-acoustic mode in edge-plasma pedestals. Phys. Rev. Lett. 100, 215,001 (2008). https://doi.org/10.1103/PhysRevLett.100.215001

    Article  Google Scholar 

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Authors and Affiliations

  1. Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, CA, 94550, USA

    Debojyoti Ghosh, Milo R. Dorr & Jeffrey A. F. Hittinger

  2. Physics Division, Lawrence Livermore National Laboratory, Livermore, CA, 94550, USA

    Mikhail A. Dorf

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  1. Debojyoti Ghosh
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  2. Mikhail A. Dorf
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  3. Milo R. Dorr
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  4. Jeffrey A. F. Hittinger
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Correspondence to Debojyoti Ghosh.

Additional information

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program.

A Computation of Rosenbluth Potentials

A Computation of Rosenbluth Potentials

The Rosenbluth potentials are related to the distribution function by Poisson equations, given by (7), that are defined on an infinite velocity domain. The algorithm implemented in COGENT to solve them on a finite numerical domain was introduced in [25] and is summarized briefly in this section. The numerical velocity domain is defined as \(\varOmega _\mathbf{v} \equiv \left[ -v_{\parallel ,\mathrm{max} },v_{\parallel ,\mathrm{max} }\right] \times \left[ 0,\mu _{\mathrm{max}}\right] \), while the infinite velocity domain is \(\varOmega _{\mathbf{v},\infty } \equiv \left[ -\infty ,\infty \right] \times \left[ 0,\infty \right] \). The Green’s function method is used to compute the boundary values as

$$\begin{aligned} \varphi \left( \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right)&= \frac{1}{4\pi } \int \limits _{\varOmega _{\mathbf{v},\infty }} \frac{f\left( \mathbf{v}'\right) }{\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}} - \mathbf{v}' \right| } d\mathbf{v}', \end{aligned}$$
(68a)
$$\begin{aligned} \rho \left( \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right)&= \frac{1}{8\pi } \int \limits _{\varOmega _{\mathbf{v},\infty }} \left| \mathbf{v}_{\partial \varOmega _\mathbf{v}} - \mathbf{v}' \right| f\left( \mathbf{v}'\right) d\mathbf{v}', \end{aligned}$$
(68b)

where \(\mathbf{v}_{\partial \varOmega _\mathbf{v}}\) is the velocity vector at the computational domain boundary \(\partial \varOmega _\mathbf{v}\). Direct evaluation of (68) is expensive and an asymptotic method is used. The Green’s function is expanded as [44]

$$\begin{aligned} \frac{1}{\left| \mathbf{v} - \mathbf{v}'\right| } = 4\pi \sum _{l=0}^{\infty } \sum _{m=-l}^{l} \frac{1}{2l+1} \frac{v_{<}^l}{v_{>}^{l+1}}Y_{lm}^*\left( \theta ',\psi '\right) Y_{lm}\left( \theta ,\psi \right) , \end{aligned}$$
(69)

where \(Y_{lm}\) is the spherical harmonic function, \(\theta = \arccos \left( v_\parallel /v\right) \) is the pitch angle, \(\psi \) is the gyro-angle, \(v_{>} = \max {\left( \left| \mathbf{v}\right| , \left| \mathbf{v}'\right| \right) }\), and \(v_{<} = \min {\left( \left| \mathbf{v}\right| , \left| \mathbf{v}'\right| \right) }\). Therefore,

$$\begin{aligned} \varphi \left( \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right) = -\frac{1}{4\pi } \sum _{l=0}^{\infty } \frac{h_l}{\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right| ^{l+1}} P_l\left( \cos {\theta }\right) ;\ \ h_l = \frac{2\pi B}{m} \int \limits _{\varOmega _\mathbf{v}} f\left( v_\parallel ,\mu \right) v^l P_l\left( \cos {\theta }\right) dv_\parallel d\mu , \end{aligned}$$
(70)

where \(P_l\) denotes the Legendre polynomials, \(v = \left| \mathbf{v}\right| \) is the particle speed, and

$$\begin{aligned} f\left( v_\parallel , \mu : v > \min {\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right| }\right) = 0. \end{aligned}$$
(71)

The second Rosenbluth potential \(\rho \) is expressed on the domain boundary in terms of the first Rosenbluth potential \(\varphi \) as

$$\begin{aligned} \rho \left( \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right)&= \frac{1}{4\pi } \int \limits _{\varOmega _{\infty }} \frac{\varphi \left( \mathbf{v}'\right) }{\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}} - \mathbf{v}' \right| } d\mathbf{v}';. \end{aligned}$$
(72)

Though this is similar to (68a), the above analysis does not apply because \(\varphi \left( v_\parallel , \mu : v > \min {\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right| }\right) \ne 0\). Equation (72) is decomposed as

$$\begin{aligned} \frac{1}{4\pi } \int \limits _{\varOmega _{\infty }} \frac{\varphi \left( \mathbf{v}'\right) }{\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}} - \mathbf{v}' \right| } d\mathbf{v}'&= \int \limits _{0}^{\min {\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right| }} v^2 dv \int \limits _{0}^{\pi } d\theta \int \limits _{0}^{2\pi } \frac{\hat{\varphi }\left( \mathbf{v}'\right) }{\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}} - \mathbf{v}' \right| } d\psi \nonumber \\&\quad + \int \limits _{0}^{\min {\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right| }} v^2 dv \int \limits _{0}^{\pi } d\theta \int \limits _{0}^{2\pi } \frac{\tilde{\varphi }\left( \mathbf{v}'\right) }{\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}} - \mathbf{v}' \right| } d\psi , \end{aligned}$$
(73)

where \(\hat{\varphi }\left( \mathbf{v}\right) \) is the numerical solution to Eq. (7a) and

$$\begin{aligned} \tilde{\varphi }\left( \mathbf{v}\right) = -\frac{1}{4\pi } \sum _{l=0}^{\infty } \frac{h_l}{\left| \mathbf{v}\right| ^{l+1}} P_l\left( \cos {\theta }\right) . \end{aligned}$$
(74)

Therefore,

$$\begin{aligned}&\int \limits _{0}^{\min {\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right| }} v^2 dv \int \limits _{0}^{\pi } d\theta \int \limits _{0}^{2\pi } \frac{\hat{\varphi }\left( \mathbf{v}'\right) }{\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}} - \mathbf{v}' \right| } d\psi = \sum _{l=0}^{\infty } \frac{g_l}{\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right| ^{l+1}}P_l\left( \cos {\theta }\right) ;\nonumber \\&\quad g_l = \frac{2\pi B}{m} \int \limits _{\varOmega _\mathbf{v}} \hat{\hat{\varphi }}\left( v_\parallel ,\mu \right) v^l P_l\left( \cos {\theta }\right) dv_\parallel d\mu , \end{aligned}$$
(75)

where \(\hat{\hat{\varphi }}\left( \mathbf{v}\right) = \hat{\varphi }\left( \mathbf{v}\right) , v \le \min {\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right| }\), \(\hat{\hat{\varphi }}\left( \mathbf{v}\right) = 0, v > \min {\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right| }\), and

$$\begin{aligned}&\int \limits _{0}^{\min {\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right| }} v^2 dv \int \limits _{0}^{\pi } d\theta \int \limits _{0}^{2\pi } \frac{\tilde{\varphi }\left( \mathbf{v}'\right) }{\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}} - \mathbf{v}' \right| } d\psi \nonumber \\&\quad = -\sum _{l=0}^{\infty } \frac{h_l P_l\left( \cos {\theta }\right) }{2l+1} \left( \frac{1}{2\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right| ^{l-1}} - \frac{1}{2\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right| ^{l+1}} + \frac{1}{\left( 2l-1\right) \left| \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right| ^{l-1}} \right) . \end{aligned}$$
(76)

The second Rosenbluth potential is thus expressed at the computational domain boundary as

$$\begin{aligned} \rho \left( \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right)&= -\frac{1}{4\pi } \sum _{l=0}^{\infty } \left[ \frac{g_l}{\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right| ^{l+1}} - \frac{h_l}{2\left( 2l+1\right) } \left( \frac{2l+1}{2l-1}\frac{}{\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right| ^{l-1}} \right. \right. \nonumber \\&\left. \left. \quad - \frac{\left\{ \min {\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right| }\right\} ^2}{\left| \mathbf{v}_{\partial \varOmega _\mathbf{v}}\right| ^{l+1}} \right) \right] P_l\left( \cos {\theta }\right) . \end{aligned}$$
(77)

Equations (70) and (77) serve as the boundary conditions for (7) on the finite numerical domain \(\varOmega _v\). Cut-cell issues, arising in evaluating (75), are solved by linear interpolations [25], and therefore, the current implementation uses second-order central finite differences to discretize the derivatives in Eq. (7). The fourth-order implementation will be investigated in the future. We use the conjugate gradient method with a structured multigrid preconditioner from the hypre library [31] to solve the discretized system of equations.

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Ghosh, D., Dorf, M.A., Dorr, M.R. et al. Kinetic Simulation of Collisional Magnetized Plasmas with Semi-implicit Time Integration. J Sci Comput 77, 819–849 (2018). https://doi.org/10.1007/s10915-018-0726-6

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