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

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.