A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit
A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit
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DOI：
10.1137/07069479x
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发表时间：
2008-10
期刊：
影响因子：
--
通讯作者：
M. Lemou;L. Mieussens
中科院分区：
文献类型：
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作者：
M. Lemou;L. Mieussens
我们提出了一种新的线性输运方程的数值格式。它是基于将分布函数分解为平衡和非平衡部分。我们还使用投影技术，使我们能够将动力学方程重新表述为宏观密度的演化方程和非平衡部分的动力学方程的耦合系统。通过适当的时间半隐式离散化，我们的方案能够精确地逼近动力学和扩散状态下的解。它在以下意义上是渐近保持的：当粒子的平均自由程较小时，我们的格式渐近等价于极限扩散模型的标准数值格式。证明了简单电报模型的一致稳定性。研究了各种边界条件。我们的方法在一维情况下通过几个数值试验和与以前的渐近保持格式的比较得到了验证。
We propose a new numerical scheme for linear transport equations. It is based on a decomposition of the distribution function into equilibrium and nonequilibrium parts. We also use a projection technique that allows us to reformulate the kinetic equation into a coupled system of an evolution equation for the macroscopic density and a kinetic equation for the nonequilibrium part. By using a suitable time semi-implicit discretization, our scheme is able to accurately approximate the solution in both kinetic and diffusion regimes. It is asymptotic preserving in the following sense: when the mean free path of the particles is small, our scheme is asymptotically equivalent to a standard numerical scheme for the limit diffusion model. A uniform stability property is proved for the simple telegraph model. Various boundary conditions are studied. Our method is validated in one-dimensional cases by several numerical tests and comparisons with previous asymptotic preserving schemes.