一种用于反应扩散问题的最小解析浸没边界模型
A Minimally-Resolved Immersed Boundary Model for Reaction-Diffusion Problems
AI总结：
本文提出一种最小解析浸没边界模型，用于模拟反应扩散过程，通过更少的自由度表示反应粒子，支持复杂形状构建，并开发多网格预处理程序以高效求解线性系统。
摘要：
我们开发了一种浸没边界方法，用于模拟反应扩散过程，特别是在反应性球形颗粒分散体中，从扩散受限到反应受限的设置。我们使用更少的自由度表示每个反应粒子的'blob'，更复杂的或更高分辨率的粒子形状可以由多个反应性blob组成。我们数值证明，在低到中等的粒子填充密度下，blob模型能提供准确的表示，其成本不显著高于在同一域内求解泊松方程。与多重展开方法不同，我们的方法不需要解析计算格林函数，而是通过标准网格离散化泊松方程来实时计算正则化的离散格林函数。这允许在实现不同边界条件、耦合流体流动或热传输以及包含其他效应如时间演化和非线性时具有高度灵活性。我们开发了基于多网格的预处理程序，用于求解使用隐式时间离散化或研究稳态时出现的线性系统。在扩散受限情况下，得到的线性系统是一个鞍点问题，其高效求解仍然是许多粒子悬浮体的挑战。我们通过与已发表的反应扩散结果在有序和无序悬浮体中的比较来验证我们的方法。
英文摘要：
We develop an immersed-boundary approach to modeling reaction-diffusion processes in dispersions of reactive spherical particles, from the diffusion-limited to the reaction-limited setting. We represent each reactive particle with a minimally-resolved 'blob' using many fewer degrees of freedom per particle than standard discretization approaches. More complicated or more highly resolved particle shapes can be built out of a collection of reactive blobs. We demonstrate numerically that the blob model can provide an accurate representation at low to moderate packing densities of the reactive particles, at a cost not much larger than solving a Poisson equation in the same domain. Unlike multipole expansion methods, our method does not require analytically-computed Green's functions, but rather, computes regularized discrete Green's functions on the fly by using a standard grid-based discretization of the Poisson equation. This allows for great flexibility in implementing different boundary conditions, coupling to fluid flow or thermal transport, and the inclusion of other effects such as temporal evolution and even nonlinearities. We develop multigrid-based preconditioners for solving the linear systems that arise when using implicit temporal discretizations or studying steady states. In the diffusion-limited case the resulting linear system is a saddle-point problem, the efficient solution of which remains a challenge for suspensions of many particles. We validate our method by comparing to published results on reaction-diffusion in ordered and disordered suspensions of reactive spheres.
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