A comparison of simulation methods with finite-difference and finite element methods for solving Vlasov-Poisson systems
We consider the one dimensional periodic Vlasov-Poisson equation and discuss various approximations. These include the particle-in-cell method, the upstream-downstream method (a finite element method) and the Lax-Wendroff method (a finite-difference method).
The model considered as a test problem numerically simulates electrons moving over a fixed, uniform positive background when, as an initial condition for the electron distribution, a Maxwellian beam is imposed. Landau damping phenomena are observed for all approximation methods.
Good agreements on charge conservation have been observed for both finite element and finite-difference methods for up to 12.5 plasma periods. Numerical experiments show that for very short plasma periods (e.g., 2 plasma periods) the total energy of a Vlasov plasma system is conserved with finite element methods; however, it is not well conserved at all for longer plasma periods. Nevertheless, with finite - difference methods energy conservation of the system satisfactorily holds for up to 10 plasma periods.
Modification of mesh sizes and time steps shows that fine mesh sizes and small time steps can be used for reducing numerical diffusion effects. Based on different numerical results, we make a comparative study of finite element and finite-difference methods with particle - in - cell methods.