A new code for electrostatic simulation by numerical integration of the Vlasov and Ampere equations using MacCormack's method

We present a new simulation code for electrostatic waves in one dimension w hich uses the Vlasov equation to integrate the distribution function and Am pere's equation to integrate the electric field forward in time. Previous V lasov codes used the Vlasov and Poisson equations. Using Ampere's equation has two advantages. First, boundary conditions do not have to be set on the electric field. Second, it forms a logical basis for an electromagnetic co de since the time integration of the electric and magnetic fields is treate d in a similar way. MacCormack's method is used to integrate the Vlasov equ ation. which was found to be easy to implement and reliable. A stability an alysis is presented for the MacCormack scheme applied to the Vlasov equatio n. Conditions for stability are more stringent than the simple Courant-Frie drich's-Lewy (CFL) conditions for the spatial and velocity grids. We provid e a simple linear function which when combined with the CFL conditions shou ld ensure stability. Simulation results for Landau damping are in excellent agreement with numerical solutions of the linear dispersion relation for a wide range of wavelengths. The code is also able to retain phase memory as demonstrated by the recurrence effect and reproduce the effects of particl e trapping. The use of Ampere's equation enables standing and traveling wav es to be produced depending on whether the current is zero or non-zero, res pectively. In simulations where the initial current is non-zero and Maxwell 's equations are satisfied initially, additional standing waves may be set up, which could be important when the coupling of wave fields from a transm itter to a plasma is considered. (C) 2001 Academic Press.