Numerical Study of the Two-Species Vlasov-Amp\`{e}re 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-Amp\`{e}re (VA) system and apply the methods to simulate current-driven ion-acoustic instability. The algorithm is generalized from our previous work for the single-species VA system and Vlasov-Maxwell (VM) system. The main feature of the schemes is their ability to preserve the total particle number and total energy on the fully discrete level regardless of mesh size. Those are desired properties of numerical schemes especially for long time simulations with under-resolved mesh. The conservation is realized by explicit and implicit energy-conserving temporal discretizations, and the discontinuous Galerkin (DG) spatial discretizations. We benchmarked 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

Energy-conserving discontinuous Galerkin methods for the Vlasov-Amp\`{e}re system

In this paper, we propose energy-conserving numerical schemes for the Vlasov-Amp\`{e}re (VA) systems. The VA system is a model used to describe the evolution of probability density function of charged particles under self consistent electric field in plasmas. It conserves many physical quantities, including the total energy which is comprised of the kinetic and electric energy. Unlike the total particle number conservation, the total energy conservation is challenging to achieve. For simulations in longer time ranges, negligence of this fact could cause unphysical results, such as plasma self heating or cooling. In this paper, we develop the first Eulerian solvers that can preserve fully discrete total energy conservation. The main components of our solvers include explicit or implicit energy-conserving temporal discretizations, an energy-conserving operator splitting for the VA equation and discontinuous Galerkin finite element methods for the spatial discretizations. We validate our schemes by rigorous derivations and benchmark numerical examples such as Landau damping, two-stream instability and bump-on-tail instability

High order operator splitting methods based on an integral deferred correction framework

Integral deferred correction (IDC) methods have been shown to be an efficient way to achieve arbitrary high order accuracy and possess good stability properties. In this paper, we construct high order operator splitting schemes using the IDC procedure to solve initial value problems (IVPs). We present analysis to show that the IDC methods can correct for both the splitting and numerical errors, lifting the order of accuracy by $r$ with each correction, where $r$ is the order of accuracy of the method used to solve the correction equation. We further apply this framework to solve partial differential equations (PDEs). Numerical examples in two dimensions of linear and nonlinear initial-boundary value problems are presented to demonstrate the performance of the proposed IDC approach.Comment: 33 pages, 22 figure