36,999 research outputs found
Asymptotic Preserving time-discretization of optimal control problems for the Goldstein-Taylor model
We consider the development of implicit-explicit time integration schemes for
optimal control problems governed by the Goldstein-Taylor model. In the
diffusive scaling this model is a hyperbolic approximation to the heat
equation. We investigate the relation of time integration schemes and the
formal Chapman-Enskog type limiting procedure. For the class of stiffly
accurate implicit-explicit Runge-Kutta methods (IMEX) the discrete optimality
system also provides a stable numerical method for optimal control problems
governed by the heat equation. Numerical examples illustrate the expected
behavior
On the Eulerian Large Eddy Simulation of disperse phase flows: an asymptotic preserving scheme for small Stokes number flows
In the present work, the Eulerian Large Eddy Simulation of dilute disperse
phase flows is investigated. By highlighting the main advantages and drawbacks
of the available approaches in the literature, a choice is made in terms of
modelling: a Fokker-Planck-like filtered kinetic equation proposed by Zaichik
et al. 2009 and a Kinetic-Based Moment Method (KBMM) based on a Gaussian
closure for the NDF proposed by Vie et al. 2014. The resulting Euler-like
system of equations is able to reproduce the dynamics of particles for small to
moderate Stokes number flows, given a LES model for the gaseous phase, and is
representative of the generic difficulties of such models. Indeed, it
encounters strong constraints in terms of numerics in the small Stokes number
limit, which can lead to a degeneracy of the accuracy of standard numerical
methods. These constraints are: 1/as the resulting sound speed is inversely
proportional to the Stokes number, it is highly CFL-constraining, and 2/the
system tends to an advection-diffusion limit equation on the number density
that has to be properly approximated by the designed scheme used for the whole
range of Stokes numbers. Then, the present work proposes a numerical scheme
that is able to handle both. Relying on the ideas introduced in a different
context by Chalons et al. 2013: a Lagrange-Projection, a relaxation formulation
and a HLLC scheme with source terms, we extend the approach to a singular flux
as well as properly handle the energy equation. The final scheme is proven to
be Asymptotic-Preserving on 1D cases comparing to either converged or
analytical solutions and can easily be extended to multidimensional
configurations, thus setting the path for realistic applications
Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems
We are interested in high-order linear multistep schemes for time
discretization of adjoint equations arising within optimal control problems.
First we consider optimal control problems for ordinary differential equations
and show loss of accuracy for Adams-Moulton and Adams-Bashford methods, whereas
BDF methods preserve high--order accuracy. Subsequently we extend these results
to semi--lagrangian discretizations of hyperbolic relaxation systems.
Computational results illustrate theoretical findings
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
A unified IMEX Runge-Kutta approach for hyperbolic systems with multiscale relaxation
In this paper we consider the development of Implicit-Explicit (IMEX)
Runge-Kutta schemes for hyperbolic systems with multiscale relaxation. In such
systems the scaling depends on an additional parameter which modifies the
nature of the asymptotic behavior which can be either hyperbolic or parabolic.
Because of the multiple scalings, standard IMEX Runge-Kutta methods for
hyperbolic systems with relaxation loose their efficiency and a different
approach should be adopted to guarantee asymptotic preservation in stiff
regimes. We show that the proposed approach is capable to capture the correct
asymptotic limit of the system independently of the scaling used. Several
numerical examples confirm our theoretical analysis
A Hybrid Godunov Method for Radiation Hydrodynamics
From a mathematical perspective, radiation hydrodynamics can be thought of as
a system of hyperbolic balance laws with dual multiscale behavior (multiscale
behavior associated with the hyperbolic wave speeds as well as multiscale
behavior associated with source term relaxation). With this outlook in mind,
this paper presents a hybrid Godunov method for one-dimensional radiation
hydrodynamics that is uniformly well behaved from the photon free streaming
(hyperbolic) limit through the weak equilibrium diffusion (parabolic) limit and
to the strong equilibrium diffusion (hyperbolic) limit. Moreover, one finds
that the technique preserves certain asymptotic limits. The method incorporates
a backward Euler upwinding scheme for the radiation energy density and flux as
well as a modified Godunov scheme for the material density, momentum density,
and energy density. The backward Euler upwinding scheme is first-order accurate
and uses an implicit HLLE flux function to temporally advance the radiation
components according to the material flow scale. The modified Godunov scheme is
second-order accurate and directly couples stiff source term effects to the
hyperbolic structure of the system of balance laws. This Godunov technique is
composed of a predictor step that is based on Duhamel's principle and a
corrector step that is based on Picard iteration. The Godunov scheme is
explicit on the material flow scale but is unsplit and fully couples matter and
radiation without invoking a diffusion-type approximation for radiation
hydrodynamics. This technique derives from earlier work by Miniati & Colella
2007. Numerical tests demonstrate that the method is stable, robust, and
accurate across various parameter regimes.Comment: accepted for publication in Journal of Computational Physics; 61
pages, 15 figures, 11 table
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