Numerical study of the two-species Vlasov–Ampère system: Energy-conserving schemes and the current-driven ion-acoustic instability

In this paper, we propose energy-conserving Eulerian solvers for the two-species Vlasov–Ampère (VA) system and apply the methods to simulate current-driven ion-acoustic instability. The two-species VA systems are of practical importance in applications, and they conserve many physical quantities including the particle number of each species and the total energy that is comprised of kinetic energy for both species and the electric energy.The main goal of this paper is to generalize our previous work for the single-species VA system [9] and Vlasov–Maxwell (VM) system [8] to the two-species case. The methodologies proposed involve careful design of temporal discretization and the use of the discontinuous Galerkin (DG) spatial discretizations. We show that the energy-conserving time discretizations for single-species equations [9] and [8] can also work for the two-species case if extended properly. Compared to other high order schemes, we emphasize that our schemes can preserve the total particle number and total energy on the fully discrete level regardless of mesh size, making them very attractive for long time simulations. We benchmark our algorithms on a test example to check the one-species limit, and the current-driven ion-acoustic instability. To simulate the current-driven ion-acoustic instability, a slight modification for the implicit method is necessary to fully decouple the split equations. This is achieved by a Gauss–Seidel type iteration technique. Numerical results verified the conservation and performance of our methods. Finally, we remark that the schemes in this paper can be readily extended to applications when the models take more general form, such as the multi-species VM equations.