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Finite Difference formulation of any lattice Boltzmann scheme

机译：任意晶格玻尔兹曼方案的有限差分公式

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摘要

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.
机译：格子玻尔兹曼方案依赖于目标问题尺寸的扩大，以便以高度可并行化和高效的类动力学方式求解偏微分方程，分为碰撞相位和流相位。尽管从计算的角度来看，这种结构具有众所周知的优势，但并不适合构建与目标方程相关的严格一致性概念，也不适合提供精确的稳定性概念。为了缓解这些不足并引入严格的框架，我们证明了任何格玻尔兹曼方案都可以在守恒变量上重写为相应的多步有限差分方案。这是通过设计基于算子、交换代数和多项式的合适形式来实现的。因此，相应的有限差分方案的一致性概念允许在线性格玻尔兹曼方案的情况下调用拉克斯-里希特迈尔定理。此外，我们发现格子玻尔兹曼方案中常用的类冯诺依曼稳定性分析与有限差分方案的冯诺依曼稳定性分析完全对应。更一般地说，用于分析有限差分方案的常用工具现在很容易用于研究格子玻尔兹曼方案。它们的相关性通过数字插图得到验证。

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