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Implicit asymptotic preserving Method for Linear Transport Equations

Communications in Computational Physics 2017年第6期22卷 157-181页

作者： Qin Li Li Wang Mathematics Department University of Wisconsin-Madison480 Lincoln Dr.MadisonWI 53705USA Departments of Mathematics and Computational Data-Enabled Science and Engineering Program State University of New York at Buffalo244 Mathematics BuildingBuffaloNY 14260USA The Optimization Group TheWisconsin Institute ofDiscoveryMadisonWI 53715USA

The computation of the radiative transfer equation is expensive mainly due to two stiff terms:the transport term and the collision *** stiffness in the former comes from the fact that particles(such as photons)travel at the speed of light,while that in the latter is due to the strong scattering in the optically thick *** study the fully implicit scheme for this equation to account for the *** main challenge in the implicit treatment is the coupling between the spacial and angular coordinates that requires the large size of the to-be-inverted matrix,which is also ill-conditioned and not necessarily *** main idea is to utilize the spectral structure of the ill-conditioned matrix to construct a pre-conditioner,which,along with an exquisite split of the spatial and angular dependence,significantly improve the condition number and allows a matrix-free *** also design a fast solver to compute this pre-conditioner explicitly in *** method is shown to be efficient in both diffusive and free streaming limit,and the computational cost is comparable to the state-of-the-art *** examples including anisotropic scattering and two-dimensional problems are provided to validate the effectiveness of our method.

关键词： Implicitmethod asymptotic preserving pre-conditioner diffusive regime free streaming limit

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An Arbitrarily High Order and asymptotic preserving Kinetic Scheme in Compressible Fluid Dynamic

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Communications on Applied Mathematics and Computation 2024年第2期6卷 963-991页

作者： Remi Abgrall Fatemeh Nassajian Mojarrad Institute of Mathematics University of ZurichWinterthurerstrasse 190CH 8057 ZürichSwitzerland

We present a class of arbitrarily high order fully explicit kinetic numerical methods in compressible fluid dynamics,both in time and space,which include the relaxation schemes by Jin and *** methods can use the CFL number larger or equal to unity on regular Cartesian meshes for the multi-dimensional *** kinetic models depend on a small parameter that can be seen as a"Knudsen"*** method is asymptotic preserving in this Knudsen ***,the computational costs of the method are of the same order of a fully explicit *** work is the extension of Abgrall et al.(2022)[3]to multidimensional *** have assessed our method on several problems for two-dimensional scalar problems and Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.

关键词： Kinetic scheme Compressible fluid dynamics High order methods Explicit schemes asymptotic preserving Defect correction method

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Spatial second-order positive and asymptotic preserving unified gas kinetic schemes for radiative transfer equations

Spatial second-order positive and asymptotic preserving unif...

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第十二届全国流体力学学术会议

作者： Xiaojing Xu Song Jiang Wenjun Sun School of Mathematics and Physics Southwest University of Science and Technology Institute of Applied Physics and Computational Mathematics Center for Applied Physics and Technology College of Engineering Peking University

We propose two spatial second-order schemes for linear radiative transfer equations by using the idea of the unified gas kinetic scheme(UGKS) to construct the numerical boundary fluxes,and show that the proposed schemes are both positive and asymptotic preserving. The UGKS was proposed by Xu and Huang [29] for continuum and rarefied flows firstly, and was then applied to a linear radiative transfer equation by Mieussens in [20] where the asymptotic preserving property of UGKS is shown. Although it is asymptotic preserving, UGKS can not always keep the positivity of solutions. We first apply UGKS to discretize a linear radiative transfer equation to have a spatial second-order scheme. Then, by a detailed analysis of the numerical boundary fluxes, we are able to find the reasons why the positive preserving property of UGKS fails. Finally, we carefully employ a linear scaling limiter and a flux correction to make UGKS positive-preserving but still asymptotic-preserving. Consequently, we propose two spatial second-order positive and asymptotic preserving unified gas kinetic schemes for the linear radiative transfer equation, thus improving the earlier work [32] where only a first-order positive and asymptotic scheme is developed. The proposed schemes can well capture the solution of the diffusion limit equation in optically thick regions without requiring the cell size being smaller than the photon’s mean free path, while the solution in optically thin regions can also be well resolved in a natural way. To our best knowledge,this is the first time that a spatial high-order(second-order) positive and asymptotic preserving gas kinetic scheme for linear radiative transfer equations is constructed. Several numerical experiments are included to validate the spatial second-order accuracy, positive-and asymptotic-preserving properties of the proposed schemes.

关键词： Radiative transfer equation unified gas kinetic scheme discrete ordinate method second-order scheme positive preserving asymptotic preserving

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A UNIFORM FIRST-ORDER METHOD FOR THE DISCRETE ORDINATE TRANSPORT EQUATION WITH INTERFACES IN X,Y-GEOMETRY

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Journal of Computational Mathematics 2009年第6期27卷 764-786页

作者： Min Tang Department of Mathematical Sciences Tsinghua University Beijing 100084 China

A uniformly first-order convergent numerical method for the discrete-ordinate transport equation in the rectangle geometry is proposed in this paper. Firstly we approximate the scattering coefficients and source terms by piecewise constants determined by their cell averages. Then for each cell, following the work of De Barros and Larsen [1, 19], the solution at the cell edge is approximated by its average along the edge. As a result, the solution of the system of equations for the cell edge averages in each cell can be obtained analytically. Finally, we piece together the numerical solution with the neighboring cells using the interface conditions. When there is no interface or boundary layer, this method is asymptotic-preserving, which implies that coarse meshes （meshes that do not resolve the mean free path） can be used to obtain good numerical approximations. Moreover, the uniform first-order convergence with respect to the mean free path is shown numerically and the rigorous proof is provided.

关键词： Transport equation Interface Diffusion limit asymptotic preserving Uniform numerical convergence X,Y-geometry.

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All Speed Scheme for the Low Mach Number Limit of the Isentropic Euler Equations

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Communications in Computational Physics 2011年第6期10卷 1-31页

作者： Pierre Degond Min Tang Institute of Mathematics of Toulouse Universite Paul Sabatier118route de Narbonne31062 Toulouse cedexFrance

An all speed scheme for the Isentropic Euler equations is presented in thispaper. When the Mach number tends to zero, the compressible Euler equations converge to their incompressible counterpart, in which the density becomes a constant. Increasing approximation errors and severe stability constraints are the main difficultyin the low Mach regime. The key idea of our all speed scheme is the special semiimplicit time discretization, in which the low Mach number stiff term is divided intotwo parts, one being treated explicitly and the other one implicitly. Moreover, the fluxof the density equation is also treated implicitly and an elliptic type equation is derivedto obtain the density. In this way, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared withprevious semi-implicit methods [11,13,29], firstly, nonphysical oscillations can be suppressed by choosing proper parameter, besides, only a linear elliptic equation needs tobe solved implicitly which reduces much computational cost. We develop this semiimplicit time discretization in the framework of a first order Local Lax-Friedrichs (orRusanov) scheme and numerical tests are displayed to demonstrate its performances.

关键词： Low Mach number Isentropic Euler equations compressible flow incompressible limit asymptotic preserving Rusanov scheme

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