74 research outputs found
Multiscale convergence properties for spectral approximations of a model kinetic equation
In this work, we prove rigorous convergence properties for a semi-discrete,
moment-based approximation of a model kinetic equation in one dimension. This
approximation is equivalent to a standard spectral method in the velocity
variable of the kinetic distribution and, as such, is accompanied by standard
algebraic estimates of the form , where is the number of modes and
depends on the regularity of the solution. However, in the multiscale
setting, the error estimate can be expressed in terms of the scaling parameter
, which measures the ratio of the mean-free-path to the
characteristic domain length. We show that, for isotropic initial conditions,
the error in the spectral approximation is . More
surprisingly, the coefficients of the expansion satisfy super convergence
properties. In particular, the error of the coefficient of the
expansion scales like when and
for all . This result is
significant, because the low-order coefficients correspond to physically
relevant quantities of the underlying system. All the above estimates involve
constants depending on , the time , and the initial condition. We
investigate specifically the dependence on , in order to assess whether
increasing actually yields an additional factor of in the error.
Numerical tests will also be presented to support the theoretical results
On Cellular Automata Models of Traffic Flow with Look-Ahead Potential
We study the statistical properties of a cellular automata model of traffic
flow with the look-ahead potential. The model defines stochastic rules for the
movement of cars on a lattice. We analyze the underlying statistical
assumptions needed for the derivation of the coarse-grained model and
demonstrate that it is possible to relax some of them to obtain an improved
coarse-grained ODE model. We also demonstrate that spatial correlations play a
crucial role in the presence of the look-ahead potential and propose a simple
empirical correction to account for the spatial dependence between neighboring
cells
A new class of high order semi-Lagrangian schemes for rarefied gas dynamics
In this paper we genealize the fast semi-Lagrangian scheme developed in [J.
Comput. Phys., Vol. 255, 2013, pp 680-698] to the case of high order
reconstructions of the distribution function. The original first order accurate
semi-Lagrangian scheme is supplemented with polynomial reconstructions of the
distribution function and of the collisional operator leading to an effective
high order accurate numerical scheme for all regimes, from extremely rarefied
gas to highly collisional siuation. The main idea relies on updating at each
time step the extreme points of the distribution function for each velocity of
the lattice instead of updating the solution in the cell centers, these
extremes points being located at different positions for any fixed velocity of
the lattice. The result is a class of scheme which permits to preserve the
structure of the solution over very long times compared to existing schemes
from the literature. We propose a proof of concept of this new approach along
with numerical tests and comparisons with classical numerical methods
A Positive Asymptotic Preserving Scheme for Linear Kinetic Transport Equations
We present a positive and asymptotic preserving numerical scheme for solving
linear kinetic, transport equations that relax to a diffusive equation in the
limit of infinite scattering. The proposed scheme is developed using a standard
spectral angular discretization and a classical micro-macro decomposition. The
three main ingredients are a semi-implicit temporal discretization, a dedicated
finite difference spatial discretization, and realizability limiters in the
angular discretization. Under mild assumptions on the initial condition and
time step, the scheme becomes a consistent numerical discretization for the
limiting diffusion equation when the scattering cross-section tends to
infinity. The scheme also preserves positivity of the particle concentration on
the space-time mesh and therefore fixes a common defect of spectral angular
discretizations. The scheme is tested on the well-known line source benchmark
problem with the usual uniform material medium as well as a medium composed
from different materials that are arranged in a checkerboard pattern. We also
report the observed order of space-time accuracy of the proposed scheme
Sparse Dynamics for Partial Differential Equations
We investigate the approximate dynamics of several differential equations
when the solutions are restricted to a sparse subset of a given basis. The
restriction is enforced at every time step by simply applying soft thresholding
to the coefficients of the basis approximation. By reducing or compressing the
information needed to represent the solution at every step, only the essential
dynamics are represented. In many cases, there are natural bases derived from
the differential equations which promote sparsity. We find that our method
successfully reduces the dynamics of convection equations, diffusion equations,
weak shocks, and vorticity equations with high frequency source terms
Realizability-Preserving DG-IMEX Method for the Two-Moment Model of Fermion Transport
Building on the framework of Zhang \& Shu
\cite{zhangShu_2010a,zhangShu_2010b}, we develop a realizability-preserving
method to simulate the transport of particles (fermions) through a background
material using a two-moment model that evolves the angular moments of a phase
space distribution function . The two-moment model is closed using algebraic
moment closures; e.g., as proposed by Cernohorsky \& Bludman
\cite{cernohorskyBludman_1994} and Banach \& Larecki
\cite{banachLarecki_2017a}. Variations of this model have recently been used to
simulate neutrino transport in nuclear astrophysics applications, including
core-collapse supernovae and compact binary mergers. We employ the
discontinuous Galerkin (DG) method for spatial discretization (in part to
capture the asymptotic diffusion limit of the model) combined with
implicit-explicit (IMEX) time integration to stably bypass short timescales
induced by frequent interactions between particles and the background.
Appropriate care is taken to ensure the method preserves strict algebraic
bounds on the evolved moments (particle density and flux) as dictated by
Pauli's exclusion principle, which demands a bounded distribution function
(i.e., ). This realizability-preserving scheme combines a suitable
CFL condition, a realizability-enforcing limiter, a closure procedure based on
Fermi-Dirac statistics, and an IMEX scheme whose stages can be written as a
convex combination of forward Euler steps combined with a backward Euler step.
Numerical results demonstrate the realizability-preserving properties of the
scheme. We also demonstrate that the use of algebraic moment closures not based
on Fermi-Dirac statistics can lead to unphysical moments in the context of
fermion transport.Comment: Submitted to Journal of Computational Physic
Implicit Filtered PN for High-Energy Density Thermal Radiation Transport using Discontinuous Galerkin Finite Elements
In this work, we provide a fully-implicit implementation of the
time-dependent, filtered spherical harmonics (FPN) equations for non-linear,
thermal radiative transfer. We investigate local filtering strategies and
analyze the effect of the filter on the conditioning of the system, showing in
particular that the filter improves the convergence properties of the iterative
solver. We also investigate numerically the rigorous error estimates derived in
the linear setting, to determine whether they hold also for the non-linear
case. Finally, we simulate a standard test problem on an unstructured mesh and
make comparisons with implicit Monte-Carlo (IMC) calculations.Comment: In this resubmission, the eigenspectrum study in the streaming limit
was removed. The interested reader is invited to look at the original
submission and/or to reference [48] of the resubmissio
Analysis of the Zero Relaxation Limit of Systems of Hyperbolic Conservation Laws with Random Initial Data
We show the convergence of the zero relaxation limit in systems of hyperbolic conservation laws with stochastic initial data. Precisely,
solutions converge to a solution of the local equilibrium approximation as the
relaxation time tends to zero. The initial data are assumed to depend on
finitely many random variables, and the convergence is then proved via the
appropriate analogues of the compensated compactness methods used in treating
the deterministic case. We also demonstrate the validity of this limit in the
case of the semi-linear -system; the well-posedness of both the system and
its equilibrium approximation are proved, and the convergence is shown with no
a priori conditions on solutions. This model serves as a prototype for
understanding how asymptotic approximations can be used as control variates for
hyperbolic balance laws with uncertainty
Diagnosing Forward Operator Error Using Optimal Transport
We investigate overdetermined linear inverse problems for which the forward
operator may not be given accurately. We introduce a new tool called the
structure, based on the Wasserstein distance, and propose the use of this to
diagnose and remedy forward operator error. Computing the structure turns out
to use an easy calculation for a Euclidean homogeneous degree one distance, the
Earth Mover's Distance, based on recently developed algorithms. The structure
is proven to distinguish between noise and signals in the residual and gives a
plan to help recover the true direct operator in some interesting cases. We
expect to use this technique not only to diagnose the error, but also to
correct it, which we do in some simple cases presented below
Hybrid Solver for the Radiative Transport Equation Using Finite Volume and Discontinuous Galerkin
We propose a hybrid spatial discretization for the radiative transport
equation that combines a second-order discontinuous Galerkin (DG) method and a
second-order finite volume (FV) method. The strategy relies on a simple
operator splitting that has been used previously to combine different angular
discretizations. Unlike standard FV methods with upwind fluxes, the hybrid
approach is able to accurately simulate problems in scattering dominated
regimes. However, it requires less memory and yields a faster computational
time than a uniform DG discretization. In addition, the underlying splitting
allows naturally for hybridization in both space and angle. Numerical results
are given to demonstrate the efficiency of the hybrid approach in the context
of discrete ordinate angular discretizations and Cartesian spatial grids.Comment: 25 pages, 5 figures, 10 table
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