58 research outputs found
A New Discontinuous Galerkin Finite Element Method for Directly Solving the Hamilton-Jacobi Equations
In this paper, we improve upon the discontinuous Galerkin (DG) method for
Hamilton-Jacobi (HJ) equation with convex Hamiltonians in (Y. Cheng and C.-W.
Shu, J. Comput. Phys. 223:398-415,2007) and develop a new DG method for
directly solving the general HJ equations. The new method avoids the
reconstruction of the solution across elements by utilizing the Roe speed at
the cell interface. Besides, we propose an entropy fix by adding penalty terms
proportional to the jump of the normal derivative of the numerical solution.
The particular form of the entropy fix was inspired by the Harten and Hyman's
entropy fix (A. Harten and J. M. Hyman. J. Comput. Phys. 50(2):235-269, 1983)
for Roe scheme for the conservation laws. The resulting scheme is compact,
simple to implement even on unstructured meshes, and is demonstrated to work
for nonconvex Hamiltonians. Benchmark numerical experiments in one dimension
and two dimensions are provided to validate the performance of the method
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
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
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