A numerical method for solving the one-dimensional Vlasov—Poisson equation in phase space

Abstract A new numerical method for solving the one-dimensional Vlasov—Poisson equation in phase space is proposed. The scheme advects the distribution function and its first derivatives in the x and v directions for one time step by using a numerical integration method for ordinary differential equations, and reconstructs the profile in phase space by using a cubic polynomial within a grid cell. The method gives stable and accurate results, and is efficient. It is successfully applied to a number of standard problems; the recurrence effect for a free streaming distribution, linear Landau damping, strong nonlinear Landau damping, the two-stream instability, and the bump-on-tail instability. A method of smoothing filamentation is given. The method can be generalized in a straightforward way to treat the Fokker—Planck equation, the Boltzmann equation, and more complicated cases such as problems with nonperoiodic boundary conditions and higher dimensional problems.

[1]  Stephen Wollman,et al.  Numerical Approximation of the One-Dimensional Vlasov--Poisson System with Periodic Boundary Conditions , 1996 .

[2]  Réal R. J. Gagné,et al.  A Splitting Scheme for the Numerical Solution of a One-Dimensional Vlasov Equation , 1977 .

[3]  Tomoaki Kunugi,et al.  Stability and accuracy of the cubic interpolated propagation scheme , 1997 .

[4]  M. Shoucri,et al.  Numerical integration of the Vlasov equation , 1974 .

[5]  Alexander J. Klimas,et al.  A numerical method based on the Fourier-Fourier transform approach for modeling 1-D electron plasma evolution. [in earth bow shock region , 1983 .

[6]  Takashi Yabe,et al.  A universal solver for hyperbolic equations by cubic-polynomial interpolation. II, Two- and three-dimensional solvers , 1991 .

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

[8]  William H. Press,et al.  Numerical recipes , 1990 .

[9]  L. Gardner,et al.  A finite element code for the simulation of one-dimensional Vlasov plasmas I. Theory , 1988 .

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

[11]  Takashi Yabe,et al.  A universal cubic interpolation solver for compressible and incompressible fluids , 1991 .

[12]  Takashi Yabe,et al.  A universal solver for hyperbolic equations by cubic-polynomial interpolation I. One-dimensional solver , 1991 .

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

[14]  L. Gardner,et al.  A finite element code for the simulation of one-dimensional Vlasov plasmas. II.Applications , 1988 .

[15]  M. Shoucri,et al.  Stability of Bernstein–Greene–Kruskal plasma equilibria. Numerical experiments over a long time , 1988 .

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

[17]  T. Yabe,et al.  Simulation technique for dynamic evaporation processes , 1995 .

[18]  Alexander J. Klimas,et al.  A method for overcoming the velocity space filamentation problem in collisionless plasma model solutions , 1987 .

[19]  M. Shoucri,et al.  BGK structures as quasi-particles , 1987 .

[20]  G. Joyce,et al.  Numerical integration methods of the Vlasov equation , 1971 .

[21]  D. Swanson,et al.  SELF-CONSISTENT FINITE AMPLITUDE WAVE DAMPING. , 1972 .

[22]  G. Knorr Plasma simulation with few particles. , 1973 .

[23]  T. Armstrong,et al.  Numerical Study of Weakly Unstable Electron Plasma Oscillations , 1969 .