论文标题
任何晶格Boltzmann方案的有限差分公式
Finite Difference formulation of any lattice Boltzmann scheme
论文作者
论文摘要
晶格Boltzmann方案依赖于目标问题大小的扩大，以便以高度可行的和高效的动力学样式求解PDE，分为碰撞和流相。尽管从计算的角度来看，尽管有众所周知的优势，但这种结构并不适合构建有关目标方程的一致性的严格概念，并提供了精确的稳定性概念。为了减轻这些短缺并引入严格的框架，我们证明了任何晶格Boltzmann方案都可以在保守变量上重写为相应的多步有限差方案。这是通过设计基于运营商，交换代数和多项式的合适形式主义来实现的。因此，相应有限差异方案的一致性概念允许在线性晶格Boltzmann方案的情况下调用Lax-Richtmyer定理。此外，我们表明，用于晶格Boltzmann方案的频繁使用的von Neumann样稳定性分析完全对应于其有限差异对应物的von Neumann稳定性分析。更普遍地，现在很容易地使用用于分析有限差异方案的通常工具来研究晶格玻尔兹曼方案。它们的相关性通过数值插图验证。
Lattice Boltzmann schemes rely on the enlargement of the size of the target problem in order to solve PDEs in a highly parallelizable and efficient kinetic-like fashion, split into a collision and a stream phase. This structure, despite the well-known advantages from a computational standpoint, is not suitable to construct a rigorous notion of consistency with respect to the target equations and to provide a precise notion of stability. In order to alleviate these shortages and introduce a rigorous framework, we demonstrate that any lattice Boltzmann scheme can be rewritten as a corresponding multi-step Finite Difference scheme on the conserved variables. This is achieved by devising a suitable formalism based on operators, commutative algebra and polynomials. Therefore, the notion of consistency of the corresponding Finite Difference scheme allows to invoke the Lax-Richtmyer theorem in the case of linear lattice Boltzmann schemes. Moreover, we show that the frequently-used von Neumann-like stability analysis for lattice Boltzmann schemes entirely corresponds to the von Neumann stability analysis of their Finite Difference counterpart. More generally, the usual tools for the analysis of Finite Difference schemes are now readily available to study lattice Boltzmann schemes. Their relevance is verified by means of numerical illustrations.
