6,657 research outputs found
Asymptotic preserving Implicit-Explicit Runge-Kutta methods for non linear kinetic equations
We discuss Implicit-Explicit (IMEX) Runge Kutta methods which are
particularly adapted to stiff kinetic equations of Boltzmann type. We consider
both the case of easy invertible collision operators and the challenging case
of Boltzmann collision operators. We give sufficient conditions in order that
such methods are asymptotic preserving and asymptotically accurate. Their
monotonicity properties are also studied. In the case of the Boltzmann
operator, the methods are based on the introduction of a penalization technique
for the collision integral. This reformulation of the collision operator
permits to construct penalized IMEX schemes which work uniformly for a wide
range of relaxation times avoiding the expensive implicit resolution of the
collision operator. Finally we show some numerical results which confirm the
theoretical analysis
Implicit-Explicit multistep methods for hyperbolic systems with multiscale relaxation
We consider the development of high order space and time numerical methods
based on Implicit-Explicit (IMEX) multistep time integrators for hyperbolic
systems with relaxation. More specifically, we consider hyperbolic balance laws
in which the convection and the source term may have very different time and
space scales. As a consequence the nature of the asymptotic limit changes
completely, passing from a hyperbolic to a parabolic system. From the
computational point of view, standard numerical methods designed for the
fluid-dynamic scaling of hyperbolic systems with relaxation present several
drawbacks and typically lose efficiency in describing the parabolic limit
regime. In this work, in the context of Implicit-Explicit linear multistep
methods we construct high order space-time discretizations which are able to
handle all the different scales and to capture the correct asymptotic behavior,
independently from its nature, without time step restrictions imposed by the
fast scales. Several numerical examples confirm the theoretical analysis
High Order Asymptotic Preserving DG-IMEX Schemes for Discrete-Velocity Kinetic Equations in a Diffusive Scaling
In this paper, we develop a family of high order asymptotic preserving
schemes for some discrete-velocity kinetic equations under a diffusive scaling,
that in the asymptotic limit lead to macroscopic models such as the heat
equation, the porous media equation, the advection-diffusion equation, and the
viscous Burgers equation. Our approach is based on the micro-macro
reformulation of the kinetic equation which involves a natural decomposition of
the equation to the equilibrium and non-equilibrium parts. To achieve high
order accuracy and uniform stability as well as to capture the correct
asymptotic limit, two new ingredients are employed in the proposed methods:
discontinuous Galerkin spatial discretization of arbitrary order of accuracy
with suitable numerical fluxes; high order globally stiffly accurate
implicit-explicit Runge-Kutta scheme in time equipped with a properly chosen
implicit-explicit strategy. Formal asymptotic analysis shows that the proposed
scheme in the limit of epsilon -> 0 is an explicit, consistent and high order
discretization for the limiting equation. Numerical results are presented to
demonstrate the stability and high order accuracy of the proposed schemes
together with their performance in the limit
Direct simulation Monte Carlo schemes for Coulomb interactions in plasmas
We consider the development of Monte Carlo schemes for molecules with Coulomb
interactions. We generalize the classic algorithms of Bird and Nanbu-Babovsky
for rarefied gas dynamics to the Coulomb case thanks to the approximation
introduced by Bobylev and Nanbu (Theory of collision algorithms for gases and
plasmas based on the Boltzmann equation and the Landau-Fokker-Planck equation,
Physical Review E, Vol. 61, 2000). Thus, instead of considering the original
Boltzmann collision operator, the schemes are constructed through the use of an
approximated Boltzmann operator. With the above choice larger time steps are
possible in simulations; moreover the expensive acceptance-rejection procedure
for collisions is avoided and every particle collides. Error analysis and
comparisons with the original Bobylev-Nanbu (BN) scheme are performed. The
numerical results show agreement with the theoretical convergence rate of the
approximated Boltzmann operator and the better performance of Bird-type schemes
with respect to the original scheme
Modeling water waves beyond perturbations
In this chapter, we illustrate the advantage of variational principles for
modeling water waves from an elementary practical viewpoint. The method is
based on a `relaxed' variational principle, i.e., on a Lagrangian involving as
many variables as possible, and imposing some suitable subordinate constraints.
This approach allows the construction of approximations without necessarily
relying on a small parameter. This is illustrated via simple examples, namely
the Serre equations in shallow water, a generalization of the Klein-Gordon
equation in deep water and how to unify these equations in arbitrary depth. The
chapter ends with a discussion and caution on how this approach should be used
in practice.Comment: 15 pages, 1 figure, 39 references. This document is a contributed
chapter to an upcoming volume to be published by Springer in Lecture Notes in
Physics Series. Other author's papers can be downloaded at
http://www.denys-dutykh.com
Space-time FLAVORS: finite difference, multisymlectic, and pseudospectral integrators for multiscale PDEs
We present a new class of integrators for stiff PDEs. These integrators are
generalizations of FLow AVeraging integratORS (FLAVORS) for stiff ODEs and SDEs
introduced in [Tao, Owhadi and Marsden 2010] with the following properties: (i)
Multiscale: they are based on flow averaging and have a computational cost
determined by mesoscopic steps in space and time instead of microscopic steps
in space and time; (ii) Versatile: the method is based on averaging the flows
of the given PDEs (which may have hidden slow and fast processes). This
bypasses the need for identifying explicitly (or numerically) the slow
variables or reduced effective PDEs; (iii) Nonintrusive: A pre-existing
numerical scheme resolving the microscopic time scale can be used as a black
box and easily turned into one of the integrators in this paper by turning the
large coefficients on over a microscopic timescale and off during a mesoscopic
timescale; (iv) Convergent over two scales: strongly over slow processes and in
the sense of measures over fast ones; (v) Structure-preserving: for stiff
Hamiltonian PDEs (possibly on manifolds), they can be made to be
multi-symplectic, symmetry-preserving (symmetries are group actions that leave
the system invariant) in all variables and variational
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
Electrokinetic and hydrodynamic properties of charged-particles systems: From small electrolyte ions to large colloids
Dynamic processes in dispersions of charged spherical particles are of
importance both in fundamental science, and in technical and bio-medical
applications. There exists a large variety of charged-particles systems,
ranging from nanometer-sized electrolyte ions to micron-sized charge-stabilized
colloids. We review recent advances in theoretical methods for the calculation
of linear transport coefficients in concentrated particulate systems, with the
focus on hydrodynamic interactions and electrokinetic effects. Considered
transport properties are the dispersion viscosity, self- and collective
diffusion coefficients, sedimentation coefficients, and electrophoretic
mobilities and conductivities of ionic particle species in an external electric
field. Advances by our group are also discussed, including a novel
mode-coupling-theory method for conduction-diffusion and viscoelastic
properties of strong electrolyte solutions. Furthermore, results are presented
for dispersions of solvent-permeable particles, and particles with non-zero
hydrodynamic surface slip. The concentration-dependent swelling of ionic
microgels is discussed, as well as a far-reaching dynamic scaling behavior
relating colloidal long- to short-time dynamics
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