<p>Kinetic transport models are mesoscopic mathematical descriptions of the transport of particles as well as their interactions with the background media or among themselves, and they have wide applications in many areas of mathematical physics such as nuclear and biomedical engineering, rarefied gas dynamics, and plasma physics. They are often multi-scale, with different characteristics (e.g. hyperbolic, diffusive) depending on the material properties. As our continuing effort in a sequence of works to design and analyze numerical methods for accurate and robust simulation of the multi-scale kinetic transport models, in this work, we consider a linear kinetic transport model, a simplified radiative transfer equation, in a diffusive scaling, and propose and analyze three families of asymptotic preserving (AP) methods. Numerical methods with the AP property, that is to preserve the asymptotic behavior of the models at the discrete level on under-resolved meshes, can work uniformly well to simulate multi-scale models across a wide range of scales. The proposed methods start from the micro-macro decomposition of the model, and involve discontinuous Galerkin methods in space, the discrete ordinates method (i.e. <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S_N$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>N</mi> </msub> </math></EquationSource> </InlineEquation> method) in velocity, and implicit-explicit (IMEX) BDF methods in time, with three different implicit-explicit partitionings. A systematic study, both analytically and computationally, is presented regarding their difference in stability, accuracy, computational complexity and AP property. These methods, with multi-step time integrators, are also compared in terms of their accuracy and efficiency with the ones that only differ in using certain IMEX Runge-Kutta (RK) methods in time. Together with our previous developments in [<CitationRef CitationID="CR17">17</CitationRef>, <CitationRef CitationID="CR18">18</CitationRef>, <CitationRef AdditionalCitationIDS="CR34" CitationID="CR33">33</CitationRef>–<CitationRef CitationID="CR35">35</CitationRef>, <CitationRef CitationID="CR39">39</CitationRef>] using IMEX-RK methods in time, the present work further contributes to high order discontinuous Galerkin AP methods for multi-scale kinetic simulation, especially by utilizing the structure of the micro-macro decomposition of the models.</p>

Multi-Scale Kinetic Simulation: Asymptotic Preserving IMEX-BDF-DG Schemes with Three Implicit-Explicit Partitionings

期刊论文
发表日期：

2026年4月25日

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Kimberly Matsuda,
Fengyan Li

摘要

Kinetic transport models are mesoscopic mathematical descriptions of the transport of particles as well as their interactions with the background media or among themselves, and they have wide applications in many areas of mathematical physics such as nuclear and biomedical engineering, rarefied gas dynamics, and plasma physics. They are often multi-scale, with different characteristics (e.g. hyperbolic, diffusive) depending on the material properties. As our continuing effort in a sequence of works to design and analyze numerical methods for accurate and robust simulation of the multi-scale kinetic transport models, in this work, we consider a linear kinetic transport model, a simplified radiative transfer equation, in a diffusive scaling, and propose and analyze three families of asymptotic preserving (AP) methods. Numerical methods with the AP property, that is to preserve the asymptotic behavior of the models at the discrete level on under-resolved meshes, can work uniformly well to simulate multi-scale models across a wide range of scales. The proposed methods start from the micro-macro decomposition of the model, and involve discontinuous Galerkin methods in space, the discrete ordinates method (i.e. $S_N$ $S_{N}$ method) in velocity, and implicit-explicit (IMEX) BDF methods in time, with three different implicit-explicit partitionings. A systematic study, both analytically and computationally, is presented regarding their difference in stability, accuracy, computational complexity and AP property. These methods, with multi-step time integrators, are also compared in terms of their accuracy and efficiency with the ones that only differ in using certain IMEX Runge-Kutta (RK) methods in time. Together with our previous developments in [17, 18, 33–35, 39] using IMEX-RK methods in time, the present work further contributes to high order discontinuous Galerkin AP methods for multi-scale kinetic simulation, especially by utilizing the structure of the micro-macro decomposition of the models.