104 research outputs found
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 with each correction,
where 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
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