A forward semi-Lagrangian method for the numerical solution of the Vlasov equation

This work deals with the numerical solution of the Vlasov equation, which provides a kinetic description of the evolution of a plasma, and is coupled with Poisson's equation. A new forward semi-Lagrangian method is developed. The distribution function is updated on a Eulerian grid, and the pseudo-particles located on the mesh nodes follow the characteristics of the equation forward for one time step, and are deposited on the 16 nearest nodes. This is an explicit way of solving the Vlasov equation on a grid of the phase space, which makes it easier to develop high-order time schemes than the backward method.

[1]  A. Staniforth,et al.  Semi-Lagrangian integration schemes for atmospheric models - A review , 1991 .

[2]  M. Kruskal,et al.  Exact Nonlinear Plasma Oscillations , 1957 .

[3]  T. Yabe,et al.  Cubic interpolated propagation scheme for solving the hyper-dimensional Vlasov-Poisson equation in phase space , 1999 .

[4]  Laurent Villard,et al.  A drift-kinetic Semi-Lagrangian 4D code for ion turbulence simulation , 2006, J. Comput. Phys..

[5]  Réal R. J. Gagné,et al.  Nonlinear behavior of a monochromatic wave in a one‐dimensional Vlasov plasma , 1978 .

[6]  Burton D. Fried,et al.  The Plasma Dispersion Function , 1961 .

[7]  Magdi Shoucri,et al.  A Hilbert-Vlasov code for the study of high-frequency plasma beatwave accelerator , 1996 .

[8]  M. Shoucri Nonlinear evolution of the bump‐on‐tail instability , 1979 .

[9]  Magdi Shoucri,et al.  Integration of the Vlasov equation along characteristics in one and two dimensions , 2003 .

[11]  G. Knorr,et al.  The integration of the vlasov equation in configuration space , 1976 .

[12]  Jacques Denavit,et al.  Numerical simulation of plasmas with periodic smoothing in phase space , 1972 .

[13]  Nigel Wood,et al.  The Parabolic Spline Method (PSM) for conservative transport problems , 2006 .

[14]  Régine Barthelmé,et al.  Conservation de la charge dans les codes PIC , 2005 .

[15]  Srinath Vadlamani,et al.  The particle-continuum method: an algorithmic unification of particle-in-cell and continuum methods , 2004, Comput. Phys. Commun..

[16]  P. H. Sakanaka,et al.  Formation of Ion‐Acoustic Collisionless Shocks , 1971 .

[17]  E. Sonnendrücker,et al.  The Semi-Lagrangian Method for the Numerical Resolution of the Vlasov Equation , 1999 .

[18]  Jeffrey S. Scroggs,et al.  A forward-trajectory global semi-Lagrangian transport scheme , 2003 .

[19]  Sebastian Reich,et al.  An explicit and conservative remapping strategy for semi‐Lagrangian advection , 2007 .

[20]  C. Birdsall,et al.  Plasma Physics via Computer Simulation , 2018 .

[21]  Nicolas Besse,et al.  Convergence of classes of high-order semi-Lagrangian schemes for the Vlasov-Poisson system , 2008, Math. Comput..

[22]  Giovanni Manfredi,et al.  Long-Time Behavior of Nonlinear Landau Damping , 1997 .

[23]  E. Sonnendrücker,et al.  Comparison of Eulerian Vlasov solvers , 2003 .

[24]  Glenn Joyce,et al.  Fourth-order poisson solver for the simulation of bounded plasmas , 1980 .

[25]  José A. Carrillo,et al.  Nonoscillatory Interpolation Methods Applied to Vlasov-Based Models , 2007, SIAM J. Sci. Comput..

[26]  Nigel Wood,et al.  A monotonic and positive–definite filter for a Semi‐Lagrangian Inherently Conserving and Efficient (SLICE) scheme , 2005 .

[27]  N Singh,et al.  Computer experiments on the formation and dynamics of electric double layers , 1980 .

[28]  Buchanan,et al.  Nonlinear electrostatic waves in collisionless plasmas. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[29]  John D. Villasenor,et al.  Rigorous charge conservation for local electromagnetic field solvers , 1992 .

[30]  S. Reich,et al.  The remapped particle‐mesh semi‐Lagrangian advection scheme , 2006, physics/0607243.

[31]  Magdi Shoucri A two-level implicit scheme for the numerical solution of the linearized vorticity equation , 1981 .

[32]  Régine Barthelmé Le problème de conservation de la charge dans le couplage des équations de Vlasov et de Maxwell , 2005 .

[33]  Nate Orlow,et al.  Hill ’ s Equation , 2010 .

[34]  P. Bertrand,et al.  Conservative numerical schemes for the Vlasov equation , 2001 .