36,376 research outputs found
The Landau-Zener transition and the surface hopping method for the 2D Dirac equation for graphene
A Lagrangian surface hopping algorithm is implemented to study the two
dimensional massless Dirac equation for Graphene with an electrostatic
potential, in the semiclassical regime. In this problem, the crossing of the
energy levels of the system at Dirac points requires a particular treatment in
the algorithm in order to describe the quantum transition-- characterized by
the Landau-Zener probability-- between different energy levels. We first derive
the Landau-Zener probability for the underlying problem, then incorporate it
into the surface hopping algorithm. We also show that different asymptotic
models for this problem derived in [O. Morandi, F. Sch{\"u}rrer, J. Phys. A:
Math. Theor. 44 (2011)] may give different transition probabilities. We conduct
numerical experiments to compare the solutions to the Dirac equation, the
surface hopping algorithm, and the asymptotic models of [O. Morandi, F.
Sch{\"u}rrer, J. Phys. A: Math. Theor. 44 (2011)]
An Asymptotic Preserving Scheme for the ES-BGK model
In this paper, we study a time discrete scheme for the initial value problem
of the ES-BGK kinetic equation. Numerically solving these equations are
challenging due to the nonlinear stiff collision (source) terms induced by
small mean free or relaxation time. We study an implicit-explicit (IMEX) time
discretization in which the convection is explicit while the relaxation term is
implicit to overcome the stiffness. We first show how the implicit relaxation
can be solved explicitly, and then prove asymptotically that this time
discretization drives the density distribution toward the local Maxwellian when
the mean free time goes to zero while the numerical time step is held fixed.
This naturally imposes an asymptotic-preserving scheme in the Euler limit. The
scheme so designed does not need any nonlinear iterative solver for the
implicit relaxation term. Moreover, it can capture the macroscopic fluid
dynamic (Euler) limit even if the small scale determined by the Knudsen number
is not numerically resolved. We also show that it is consistent to the
compressible Navier-Stokes equations if the viscosity and heat conductivity are
numerically resolved. Several numerical examples, in both one and two space
dimensions, are used to demonstrate the desired behavior of this scheme
Local sensitivity analysis for the Cucker-Smale model with random inputs
We present pathwise flocking dynamics and local sensitivity analysis for the
Cucker-Smale(C-S) model with random communications and initial data. For the
deterministic communications, it is well known that the C-S model can model
emergent local and global flocking dynamics depending on initial data and
integrability of communication function. However, the communication mechanism
between agents are not a priori clear and needs to be figured out from observed
phenomena and data. Thus, uncertainty in communication is an intrinsic
component in the flocking modeling of the C-S model. In this paper, we provide
a class of admissible random uncertainties which allows us to perform the local
sensitivity analysis for flocking and establish stability to the random C-S
model with uncertain communication.Comment: 32 page
A High Order Stochastic Asymptotic Preserving Scheme for Chemotaxis Kinetic Models with Random Inputs
In this paper, we develop a stochastic Asymptotic-Preserving (sAP) scheme for
the kinetic chemotaxis system with random inputs, which will converge to the
modified Keller-Segel model with random inputs in the diffusive regime. Based
on the generalized Polynomial Chaos (gPC) approach, we design a high order
stochastic Galerkin method using implicit-explicit (IMEX) Runge-Kutta (RK) time
discretization with a macroscopic penalty term. The new schemes improve the
parabolic CFL condition to a hyperbolic type when the mean free path is small,
which shows significant efficiency especially in uncertainty quantification
(UQ) with multi-scale problems. The stochastic Asymptotic-Preserving property
will be shown asymptotically and verified numerically in several tests. Many
other numerical tests are conducted to explore the effect of the randomness in
the kinetic system, in the aim of providing more intuitions for the theoretic
study of the chemotaxis models
- …