A New Code for Electrostatic Simulation by Numerical Integration of the Vlasov and Ampère Equations Using MacCormack's Method

Abstract

We present a new simulation code for electrostatic waves in one dimension which uses the Vlasov equation to integrate the distribution function and Ampère's equation to integrate the electric field forward in time. Previous Vlasov codes used the Vlasov and Poisson equations. Using Ampère'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 code since the time integration of the electric and magnetic fields is treated in a similar way. MacCormack's method is used to integrate the Vlasov equation, which was found to be easy to implement and reliable. A stability analysis is presented for the MacCormack scheme applied to the Vlasov equation. Conditions for stability are more stringent than the simple Courant–Friedrich's–Lewy (CFL) conditions for the spatial and velocity grids. We provide a simple linear function which when combined with the CFL conditions should 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 particle trapping. The use of Ampère's equation enables standing and traveling waves to be produced depending on whether the current is zero or non-zero, respectively. 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 transmitter to a plasma is considered.