We introduce and study a notion of Asymptotic Preserving schemes, related to convergence in distribution, for a class of slow-fast Stochastic Differential Equations. In some examples, crude schemes fail to capture the correct limiting equation resulting from averaging and diffusion approximation procedures. We propose examples of Asymptotic Preserving schemes: when the time-scale separation vanishes, one obtains a limiting scheme, which is shown to be consistent in distribution with the limiting Stochastic Differential Equation. Numerical experiments illustrate the importance of the proposed Asymptotic Preserving schemes for several examples. In addition, in the averaging regime, error estimates are obtained and the proposed scheme is proved to be uniformly accurate.
We design numerical schemes for a class of slow-fast systems of stochastic differential equations, where the fast component is an Ornstein-Uhlenbeck process and the slow component is driven by a fractional Brownian motion with Hurst index $H>1/2$. We establish the asymptotic preserving property of the proposed scheme: when the time-scale parameter goes to $0$, a limiting scheme which is consistent with the averaged equation is obtained. With this numerical analysis point of view, we thus illustrate the recently proved averaging result for the considered SDE systems and the main differences wit…
We consider the problem of coarse-graining in the context of finite-volume fluid models. If a variable is defined on a high-resolution grid it may be coarse-grained so that it is defined on a grid of lower resolution. In general this will cause some information about the variable to be lost. In particular, horizontal divergences, gradients or other operators calculated on the coarse grid after projecting may differ from those calculated on the fine grid. In some cases we are able to choose averaging weights for coarse-graining such that the coarse-grid operators will give a result approximatin…
We develop efficient asymptotic-preserving time discretization schemes to solve the disparate mass kinetic system of a binary gas or plasma in the "relaxation time scale" relevant to the epochal relaxation phenomenon. Since the resulting model is associated to a parameter given by the square of the mass ratio between the light and heavy particles, we develop an asymptotic-preserving scheme in a novel decomposition strategy using the penalization method for multiscale collisional kinetic equations. Both the Boltzmann and Fokker-Planck-Landau (FPL) binary collision operators will be considered. …
We present the asymptotic transitions from microscopic to macroscopic physics, their computational challenges and the Asymptotic-Preserving (AP) strategies to efficiently compute multiscale physical problems. Specifically, we will first study the asymptotic transition from quantum to classical mechanics, from classical mechanics to kinetic theory, and then from kinetic theory to hydrodynamics. We then review some representative AP schemes that mimic, at the discrete level, these asymptotic transitions, hence can be used crossing scales and, in particular, capture the macroscopic behavior witho…
In this paper, we propose to combine the fifth order Hermite weighted essentially non-oscillatory (HWENO) scheme and fast sweeping method (FSM) for the solution of the steady-state $S_{N}$ transport equation in the finite volume framework. It is well-known that the $S_{N}$ transport equation asymptotically converges to a macroscopic diffusion equation in the limit of optically thick systems with small absorption and sources. Numerical methods which can preserve the asymptotic limit are referred to as asymptotic preserving methods. In the one-dimensional case, we provide the analysis to demonst…
The present paper is devoted to the convergence analysis of a class of asymptotic preserving particle schemes [Filbet \& Rodrigues, SIAM J. Numer. Anal., 54 (2) (2016)] for the Vlasov equation with a strong external magnetic field. In this regime, classical Particle-In-Cell (PIC) methods are subject to quite restrictive stability constraints on the time and space steps, due to the small Larmor radius and plasma frequency. The asymptotic preserving discretization that we are going to study removes such a constraint while capturing the large-scale dynamics, even when the discretization (in time …
In this short paper, we intend to describe one way to construct arbitrarily high order kinetic schemes on regular meshes. The method can be arbitrarily high order in space and time, run at least CFL one, is asymptotic preserving and computationally explicit, i.e., the computational costs are of the same order of a fully explicit scheme. We also introduce a non linear stability method that enables to simulate problems with discontinuities, and it does not kill the accuracy for smooth regular solutions.
In this paper, high order semi-implicit well-balanced and asymptotic preserving finite difference WENO schemes are proposed for the shallow water equations with a non-flat bottom topography. We consider the Froude number ranging from O(1) to 0, which in the zero Froude limit becomes the "lake equations" for balanced flow without gravity waves. We apply a well-balanced finite difference WENO reconstruction, coupled with a stiffly accurate implicit-explicit (IMEX) Runge-Kutta time discretization. The resulting semi-implicit scheme can be shown to be well-balanced, asymptotic preserving (AP) and …
In this paper, we propose a high order semi-implicit well-balanced finite difference scheme for all Mach Euler equations with a gravitational source term. To obtain the asymptotic preserving property, we start from the conservative form of full compressible Euler equations and add the evolution equation of the perturbation of potential temperature. The resulting system is then split into a (non-stiff) nonlinear low dynamic material wave to be treated explicitly, and (stiff) fast acoustic and gravity waves to be treated implicitly. With the aid of explicit time evolution for the perturbation of…
We consider coupled models for particulate flows, where the disperse phase is made of particles with distinct sizes. We are thus led to a system coupling the incompressible Navier-Stokes equations to the multi-component Vlasov-Fokker-Planck equations. We design an asymptotic-preserving numerical scheme to approximate the system. The scheme is based on suitable implicit treatment of the stiff drag force term as well as the Fokker-Planck operator, and can be formally shown to capture the hydrodynamic limit with time step and mesh size independent of the Stokes number. Numerical examples illustra…
Asymptotic preserving schemes on distorted meshes for Friedrichs systems with sti relaxation: application to angular models in linear transport
CEA, DAM, DIF（法国原子能和替代能源委员会、国防事务局、信息与通信技术部）；Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie（雅克-路易·利翁斯实验室，皮埃尔和玛丽·居里大学）；Max Planck Institute for Plasma Physics（马克斯·普朗克等离子体物理研究所）
In this paper, a new asymptotic preserving (AP) scheme is proposed for the anisotropic elliptic equations. Different from previous AP schemes, the actual one is based on first-order system least-squares for second-order partial differential equations, and it is uniformly well-posed with respect to anisotropic strength. The numerical computation is realized by a deep neural network (DNN), where least-squares functionals are employed as loss functions to determine parameters of DNN. Numerical results show that the current AP scheme is easy for implementation and is robust to approximate solution…
High order asymptotic-preserving schemes for the Boltzmann equation
Université de Toulouse（图卢兹大学；法国国家科学研究中心数学研究所）；UPS, INSA, UT1, UTM（费拉拉大学数学系和CMCS）；CNRS, UMR 5219；Institut de Mathématiques de Toulouse；Mathematics Department, University of Ferrara and CMCS
