280 research outputs found
Kinetic Solvers with Adaptive Mesh in Phase Space
An Adaptive Mesh in Phase Space (AMPS) methodology has been developed for
solving multi-dimensional kinetic equations by the discrete velocity method. A
Cartesian mesh for both configuration (r) and velocity (v) spaces is produced
using a tree of trees data structure. The mesh in r-space is automatically
generated around embedded boundaries and dynamically adapted to local solution
properties. The mesh in v-space is created on-the-fly for each cell in r-space.
Mappings between neighboring v-space trees implemented for the advection
operator in configuration space. We have developed new algorithms for solving
the full Boltzmann and linear Boltzmann equations with AMPS. Several recent
innovations were used to calculate the discrete Boltzmann collision integral
with dynamically adaptive mesh in velocity space: importance sampling,
multi-point projection method, and the variance reduction method. We have
developed an efficient algorithm for calculating the linear Boltzmann collision
integral for elastic and inelastic collisions in a Lorentz gas. New AMPS
technique has been demonstrated for simulations of hypersonic rarefied gas
flows, ion and electron kinetics in weakly ionized plasma, radiation and light
particle transport through thin films, and electron streaming in
semiconductors. We have shown that AMPS allows minimizing the number of cells
in phase space to reduce computational cost and memory usage for solving
challenging kinetic problems
High-order Discretization of a Gyrokinetic Vlasov Model in Edge Plasma Geometry
We present a high-order spatial discretization of a continuum gyrokinetic
Vlasov model in axisymmetric tokamak edge plasma geometries. Such models
describe the phase space advection of plasma species distribution functions in
the absence of collisions. The gyrokinetic model is posed in a four-dimensional
phase space, upon which a grid is imposed when discretized. To mitigate the
computational cost associated with high-dimensional grids, we employ a
high-order discretization to reduce the grid size needed to achieve a given
level of accuracy relative to lower-order methods. Strong anisotropy induced by
the magnetic field motivates the use of mapped coordinate grids aligned with
magnetic flux surfaces. The natural partitioning of the edge geometry by the
separatrix between the closed and open field line regions leads to the
consideration of multiple mapped blocks, in what is known as a mapped
multiblock (MMB) approach. We describe the specialization of a more general
formalism that we have developed for the construction of high-order,
finite-volume discretizations on MMB grids, yielding the accurate evaluation of
the gyrokinetic Vlasov operator, the metric factors resulting from the MMB
coordinate mappings, and the interaction of blocks at adjacent boundaries. Our
conservative formulation of the gyrokinetic Vlasov model incorporates the fact
that the phase space velocity has zero divergence, which must be preserved
discretely to avoid truncation error accumulation. We describe an approach for
the discrete evaluation of the gyrokinetic phase space velocity that preserves
the divergence-free property to machine precision
A 4th-Order Particle-in-Cell Method with Phase-Space Remapping for the Vlasov-Poisson Equation
Numerical solutions to the Vlasov-Poisson system of equations have important
applications to both plasma physics and cosmology. In this paper, we present a
new Particle-in-Cell (PIC) method for solving this system that is 4th-order
accurate in both space and time. Our method is a high-order extension of one
presented previously [B. Wang, G. Miller, and P. Colella, SIAM J. Sci. Comput.,
33 (2011), pp. 3509--3537]. It treats all of the stages of the standard PIC
update - charge deposition, force interpolation, the field solve, and the
particle push - with 4th-order accuracy, and includes a 6th-order accurate
phase-space remapping step for controlling particle noise. We demonstrate the
convergence of our method on a series of one- and two- dimensional
electrostatic plasma test problems, comparing its accuracy to that of a
2nd-order method. As expected, the 4th-order method can achieve comparable
accuracy to the 2nd-order method with many fewer resolution elements.Comment: 18 pages, 10 figures, submitted to SIS
Robustness of Cosmological Simulations I: Large Scale Structure
The gravitationally-driven evolution of cold dark matter dominates the
formation of structure in the Universe over a wide range of length scales.
While the longest scales can be treated by perturbation theory, a fully
quantitative understanding of nonlinear effects requires the application of
large-scale particle simulation methods. Additionally, precision predictions
for next-generation observations, such as weak gravitational lensing, can only
be obtained from numerical simulations. In this paper, we compare results from
several N-body codes using test problems and a diverse set of diagnostics,
focusing on a medium resolution regime appropriate for studying many
observationally relevant aspects of structure formation. Our conclusions are
that -- despite the use of different algorithms and error-control methodologies
-- overall, the codes yield consistent results. The agreement over a wide range
of scales for the cosmological tests is test-dependent. In the best cases, it
is at the 5% level or better, however, for other cases it can be significantly
larger than 10%. These include the halo mass function at low masses and the
mass power spectrum at small scales. While there exist explanations for most of
the discrepancies, our results point to the need for significant improvement in
N-body errors and their understanding to match the precision of near-future
observations. The simulation results, including halo catalogs, and initial
conditions used, are publicly available.Comment: 32 pages, 53 figures, data from the simulations is available at
http://t8web.lanl.gov/people/heitmann/arxiv, accepted for publication in
ApJS, several minor revisions, reference added, main conclusions unchange
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