11 research outputs found
A High Order Multi-Dimensional Characteristic Tracing Strategy for the Vlasov-Poisson System
In this paper, we consider a finite difference grid-based semi-Lagrangian
approach in solving the Vlasov-Poisson (VP) system. Many of existing methods
are based on dimensional splitting, which decouples the problem into solving
linear advection problems, see {\em Cheng and Knorr, Journal of Computational
Physics, 22(1976)}. However, such splitting is subject to the splitting error.
If we consider multi-dimensional problems without splitting, difficulty arises
in tracing characteristics with high order accuracy. Specifically, the
evolution of characteristics is subject to the electric field which is
determined globally from the distribution of particle densities via the
Poisson's equation. In this paper, we propose a novel strategy of tracing
characteristics high order in time via a two-stage multi-derivative
prediction-correction approach and by using moment equations of the VP system.
With the foot of characteristics being accurately located, we proposed to use
weighted essentially non-oscillatory (WENO) interpolation to recover function
values between grid points, therefore to update solutions at the next time
level. The proposed algorithm does not have time step restriction as Eulerian
approach and enjoys high order spatial and temporal accuracy. However, such
finite difference algorithm does not enjoy mass conservation; we discuss one
possible way of resolving such issue and its potential challenge in numerical
stability. The performance of the proposed schemes are numerically demonstrated
via classical test problems such as Landau damping and two stream
instabilities
A sparse grid discontinuous Galerkin method for high-dimensional transport equations and its application to kinetic simulations
In this paper, we develop a sparse grid discontinuous Galerkin (DG) scheme
for transport equations and applied it to kinetic simulations. The method uses
the weak formulations of traditional Runge-Kutta DG (RKDG) schemes for
hyperbolic problems and is proven to be stable and convergent. A major
advantage of the scheme lies in its low computational and storage cost due to
the employed sparse finite element approximation space. This attractive feature
is explored in simulating Vlasov and Boltzmann transport equations. Good
performance in accuracy and conservation is verified by numerical tests in up
to four dimensions
Stochastic Galerkin Methods for the Boltzmann-Poisson system
We study uncertainty quantification for a Boltzmann-Poisson system that
models electron transport in semiconductors and the physical collision
mechanisms over the charges. We use the stochastic Galerkin method in order to
handle the randomness associated with the problem. The main uncertainty in the
Boltzmann equation concerns the initial conditions for a large number of
particles, which is why the problem is formulated in terms of a probability
density in phase space. The second source of uncertainty, directly related to
the quantum nature of the problem, is the collision operator, as its structure
in this semiclassical model comes from the quantum scattering matrices
operating on the wave function associated to the electron probability density.
Additional sources of uncertainty are transport, boundary data, etc. In this
study we choose first the phonon energy as a random variable, since its value
influences the energy jump appearing in the collision integral for
electron-phonon scattering. Then we choose the lattice temperature as a random
variable, since it defines the value of the collision operator terms in the
case of electron-phonon scattering by being a parameter of the phonon
distribution. The random variable for this case is a scalar then. Finally, we
present our numerical simulations.Comment: A few style corrections were performed. Extension of abstract and
conclusions in preprin
Study of conservation and recurrence of Runge-Kutta discontinuous Galerkin schemes for Vlasov-Poisson systems
In this paper we consider Runge-Kutta discontinuous Galerkin (RKDG) schemes
for Vlasov-Poisson systems that model collisionless plasmas. One-dimensional
systems are emphasized. The RKDG method, originally devised to solve
conservation laws, is seen to have excellent conservation properties, be
readily designed for arbitrary order of accuracy, and capable of being used
with a positivity-preserving limiter that guarantees positivity of the
distribution functions. The RKDG solver for the Vlasov equation is the main
focus, while the electric field is obtained through the classical
representation by Green's function for the Poisson equation. A rigorous study
of recurrence of the DG methods is presented by Fourier analysis, and the
impact of different polynomial spaces and the positivity-preserving limiters on
the quality of the solutions is ascertained. Several benchmark test problems,
such as Landau damping, two-stream instability and the KEEN (Kinetic
Electrostatic Electron Nonlinear) wave, are given
A Conservative Scheme for Vlasov Poisson Landau modeling collisional plasmas
We have developed a deterministic conservative solver for the inhomogeneous
Fokker-Planck-Landau equation coupled with the Poisson equation, which is a
{classical mean-field} primary model for collisional plasmas. Two subproblems,
i.e. the Vlasov-Poisson problem and homogeneous Landau problem, are obtained
through time-splitting methods, and treated separately by the Runge-Kutta
Discontinuous Galerkin method and a conservative spectral method, respectively.
To ensure conservation when projecting between the two different computing
grids, a special conservation routine is designed to link the solutions of
these two subproblems. This conservation routine accurately enforces
conservation of moments in Fourier space. The entire numerical scheme is
implemented with parallelization with hybrid MPI and OpenMP. Numerical
experiments are provided to study linear and nonlinear Landau Damping problems
and two-stream flow problem as well.Comment: To appear in Jour. Comp. Physic
Discontinuous Galerkin Deterministic Solvers for a Boltzmann-Poisson Model of Hot Electron Transport by Averaged Empirical Pseudopotential Band Structures
The purpose of this work is to incorporate numerically, in a discontinuous
Galerkin (DG) solver of a Boltzmann-Poisson model for hot electron transport,
an electronic conduction band whose values are obtained by the spherical
averaging of the full band structure given by a local empirical pseudopotential
method (EPM) around a local minimum of the conduction band for silicon, as a
midpoint between a radial band model and an anisotropic full band, in order to
provide a more accurate physical description of the electron group velocity and
conduction energy band structure in a semiconductor. This gives a better
quantitative description of the transport and collision phenomena that
fundamentally define the behaviour of the Boltzmann - Poisson model for
electron transport used in this work. The numerical values of the derivatives
of this conduction energy band, needed for the description of the electron
group velocity, are obtained by means of a cubic spline interpolation. The
EPM-Boltzmann-Poisson transport with this spherically averaged EPM calculated
energy surface is numerically simulated and compared to the output of
traditional analytic band models such as the parabolic and Kane bands,
numerically implemented too, for the case of 1D silicon diodes with
400nm and 50nm channels. Quantitative differences are observed in the kinetic
moments related to the conduction energy band used, such as mean velocity,
average energy, and electric current (momentum).Comment: submission to CMAME (Computer Methods in Applied Mechanics and
Engineering) Journal as a reply to the reviewers on February 201
Galerkin Methods for Boltzmann-Poisson transport with reflection conditions on rough boundaries
We consider in this paper the mathematical and numerical modelling of
reflective boundary conditions (BC) associated to Boltzmann - Poisson systems,
including diffusive reflection in addition to specularity, in the context of
electron transport in semiconductor device modelling at nano scales, and their
implementation in Discontinuous Galerkin (DG) schemes. We study these BC on the
physical boundaries of the device and develop a numerical approximation to
model an insulating boundary condition, or equivalently, a pointwise zero flux
mathematical condition for the electron transport equation. Such condition
balances the incident and reflective momentum flux at the microscopic level,
pointwise at the boundary, in the case of a more general mixed reflection with
momentum dependant specularity probability . We compare the
computational prediction of physical observables given by the numerical
implementation of these different reflection conditions in our DG scheme for BP
models, and observe that the diffusive condition influences the kinetic moments
over the whole domain in position space.Comment: Paper accepted for publication in Journal of Computational Physics.
-Conclusions section expanded -Title changed with respect to previous
preprint version -New subsections related to simulations of 2D double gated
MOSFET and comparison of bulk silicon with collisionless plasma under
reflective and periodic boundary condition
Entropy-stable positivity-preserving DG schemes for Boltzmann-Poisson models of collisional electronic transport along energy bands
This work develops entropy-stable positivity-preserving DG methods as a
computational scheme for Boltzmann-Poisson systems modeling the pdf of
electronic transport along energy bands in semiconductor crystal lattices. We
pose, using spherical or energy-angular variables as momentum coordinates, the
corresponding Vlasov Boltzmann eq. with a linear collision operator with a
singular measure modeling the scattering as functions of the energy band. We
show stability results of semi-discrete DG schemes under an entropy norm for
1D-position 2D-momentum, and 2D-position 3D-momentum, using the dissipative
properties of the collisional operator given its entropy inequality, which
depends on the whole Hamiltonian rather than only the kinetic energy. For the
1D problem, knowledge of the analytic solution to Poisson and of the
convergence to a constant current is crucial to obtain full stability. For the
2D problem, specular reflection BC are considered in addition to periodicity in
the estimate for stability under an entropy norm. Regarding positivity
preservation (1D position), we treat the collision operator as a source term
and find convex combinations of the transport and collision terms which
guarantee the positivity of the cell average of our numerical pdf at the next
time step. The positivity of the numerical pdf in the whole domain is
guaranteed by applying the natural limiters that preserve the cell average but
modify the slope of the piecewise linear solutions in order to make the
function non-negative. The use of a spherical coordinate system
is slightly different to the choice
in previous DG solvers for BP, since the proposed DG formulation gives simpler
integrals involving just piecewise polynomial functions for both transport and
collision terms, which is more adequate for Gaussian quadrature than previous
approaches.Comment: Preprint. The first author acknowledges discussions with Eirik Endeve
and Cory Hauck during the time he spent visiting them at ORNL (restricted to
positivity preservation, appearing on arXiv:1711.03949) on a trip paid by the
kinet grant NSF-RNMS DMS-1107465. V2 is a replacement for V1. V2 has added
some references and corrected typos (not affecting any of the calculations
Fast sparse grid simulations of fifth order WENO scheme for high dimensional hyperbolic PDEs
The weighted essentially non-oscillatory (WENO) schemes are a popular class
of high order accurate numerical methods for solving hyperbolic partial
differential equations (PDEs). However when the spatial dimensions are high,
the number of spatial grid points increases significantly. It leads to large
amount of operations and computational costs in the numerical simulations by
using nonlinear high order accuracy WENO schemes such as a fifth order WENO
scheme. How to achieve fast simulations by high order WENO methods for high
spatial dimension hyperbolic PDEs is a challenging and important question. In
the literature, sparse-grid technique has been developed as a very efficient
approximation tool for high dimensional problems. In a recent work [Lu, Chen
and Zhang, Pure and Applied Mathematics Quarterly, 14 (2018) 57-86], a third
order finite difference WENO method with sparse-grid combination technique was
designed to solve multidimensional hyperbolic equations including both linear
advection equations and nonlinear Burgers' equations. In application problems,
higher than third order WENO schemes are often preferred in order to
efficiently resolve the complex solution structures. In this paper, we extend
the approach to higher order WENO simulations specifically the fifth order WENO
scheme. A fifth order WENO interpolation is applied in the prolongation part of
the sparse-grid combination technique to deal with discontinuous solutions.
Benchmark problems are first solved to show that significant CPU times are
saved while both fifth order accuracy and stability of the WENO scheme are
preserved for simulations on sparse grids. The fifth order sparse grid WENO
method is then applied to kinetic problems modeled by high dimensional Vlasov
based PDEs to further demonstrate large savings of computational costs by
comparing with simulations on regular single grids.Comment: 36 pages. arXiv admin note: text overlap with arXiv:1804.0072
Bound-Preserving Discontinuous Galerkin Methods for Conservative Phase Space Advection in Curvilinear Coordinates
We extend the positivity-preserving method of Zhang & Shu (2010, JCP, 229,
3091-3120) to simulate the advection of neutral particles in phase space using
curvilinear coordinates. The ability to utilize these coordinates is important
for non-equilibrium transport problems in general relativity and also in
science and engineering applications with specific geometries. The method
achieves high-order accuracy using Discontinuous Galerkin (DG) discretization
of phase space and strong stability-preserving, Runge-Kutta (SSP-RK) time
integration. Special care in taken to ensure that the method preserves strict
bounds for the phase space distribution function ; i.e., . The
combination of suitable CFL conditions and the use of the high-order limiter
proposed in Zhang & Shu (2010) is sufficient to ensure positivity of the
distribution function. However, to ensure that the distribution function
satisfies the upper bound, the discretization must, in addition, preserve the
divergence-free property of the phase space flow. Proofs that highlight the
necessary conditions are presented for general curvilinear coordinates, and the
details of these conditions are worked out for some commonly used coordinate
systems (i.e., spherical polar spatial coordinates in spherical symmetry and
cylindrical spatial coordinates in axial symmetry, both with spherical momentum
coordinates). Results from numerical experiments --- including one example in
spherical symmetry adopting the Schwarzschild metric --- demonstrate that the
method achieves high-order accuracy and that the distribution function
satisfies the maximum principle.Comment: Submitted to JC