Analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit

We present a mathematical analysis of the asymptotic preserving scheme proposed in [M. Lemou and L. Mieussens, SIAM J. Sci. Comput., 31, pp. 334-368, 2008] for linear transport equations in kinetic and diffusive regimes. We prove that the scheme is uniformly stable and accurate with respect to the mean free path of the particles. This property is satisfied under an explicitly given CFL condition. This condition tends to a parabolic CFL condition for small mean free paths, and is close to a convection CFL condition for large mean free paths. Ou r analysis is based on very simple energy estimates

Time--Splitting Schemes and Measure Source Terms for a Quasilinear Relaxing System

Several singular limits are investigated in the context of a $2 \times 2$ system arising for instance in the modeling of chromatographic processes. In particular, we focus on the case where the relaxation term and a $L^2$ projection operator are concentrated on a discrete lattice by means of Dirac measures. This formulation allows to study more easily some time-splitting numerical schemes

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

A radiation-hydrodynamics scheme valid from the transport to the diffusion limit

We present in this paper the numerical treatment of the coupling between hydrodynamics and radiative transfer. The fluid is modeled by classical conservation laws (mass, momentum and energy) and the radiation by the grey moment $M_1$ system. The scheme introduced is able to compute accurate numerical solution over a broad class of regimes from the transport to the diffusive limits. We propose an asymptotic preserving modification of the HLLE scheme in order to treat correctly the diffusion limit. Several numerical results are presented, which show that this approach is robust and have the correct behavior in both the diffusive and free-streaming limits. In the last numerical example we test this approach on a complex physical case by considering the collapse of a gas cloud leading to a proto-stellar structure which, among other features, exhibits very steep opacity gradients.Comment: 29 pages, submitted to Journal of Computational physic

Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit

We investigate a projective integration scheme for a kinetic equation in the limit of vanishing mean free path, in which the kinetic description approaches a diffusion phenomenon. The scheme first takes a few small steps with a simple, explicit method, such as a spatial centered flux/forward Euler time integration, and subsequently projects the results forward in time over a large time step on the diffusion time scale. We show that, with an appropriate choice of the inner step size, the time-step restriction on the outer time step is similar to the stability condition for the diffusion equation, whereas the required number of inner steps does not depend on the mean free path. We also provide a consistency result. The presented method is asymptotic-preserving, in the sense that the method converges to a standard finite volume scheme for the diffusion equation in the limit of vanishing mean free path. The analysis is illustrated with numerical results, and we present an application to the Su-Olson test

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

Numerical methods for one-dimensional aggregation equations

We focus in this work on the numerical discretization of the one dimensional aggregation equation \pa_t\rho + \pa_x (v\rho)=0, $v=a(W'*\rho)$, in the attractive case. Finite time blow up of smooth initial data occurs for potential $W$ having a Lipschitz singularity at the origin. A numerical discretization is proposed for which the convergence towards duality solutions of the aggregation equation is proved. It relies on a careful choice of the discretized macroscopic velocity $v$ in order to give a sense to the product $v \rho$. Moreover, using the same idea, we propose an asymptotic preserving scheme for a kinetic system in hyperbolic scaling converging towards the aggregation equation in hydrodynamical limit. Finally numerical simulations are provided to illustrate the results