从格子Boltzmann到多步有限差分方法的初始化：修正方程与离散可观测性
Initialisation from lattice Boltzmann to multi-step Finite Difference methods: modified equations and discrete observability
AI总结：
本文通过修正方程分析格子Boltzmann方法的初始化问题，揭示初始化与主体方法修正方程匹配项数越多则数值解越光滑，并利用离散可观测性缺失将复杂约束简化为易于研究的条件。
摘要：
初始化选择的自由度是一步扩展状态空间方法和多步方法之间的共同特征。本文聚焦于格子Boltzmann格式，它们可以被解释为上述两类数值格式的例子。我们提出对格子Boltzmann方法的初始化格式进行修正方程分析，该初始化由初始数据的选择决定。这些修正方程为设计和分析初始化提供了指导，涉及相对于目标Cauchy问题的一致性阶数以及数值解的时间光滑性。具体而言，初始化与主体方法的修正方程之间匹配的项数越多，所得数值解就越光滑。这在数值耗散方面表现得尤为明显。从实现时间光滑性的约束条件出发——由于必须考虑寄生模式，这些约束可能很快变得难以处理——我们解释了某些格子Boltzmann格式（被视为交换环上的动力系统）明显缺乏可观测性如何能够产生相当简单的条件，并且就其初始化而言容易被研究。这源于全离散层面上初始化格式数量的减少。这些理论结果在若干格子Boltzmann方法上得到了成功验证。
英文摘要：
Latitude on the choice of initialisation is a shared feature between one-step extended state-space and multi-step methods. The paper focuses on lattice Boltzmann schemes, which can be interpreted as examples of both previous categories of numerical schemes. We propose a modified equation analysis of the initialisation schemes for lattice Boltzmann methods, determined by the choice of initial data. These modified equations provide guidelines to devise and analyze the initialisation in terms of order of consistency with respect to the target Cauchy problem and time smoothness of the numerical solution. In detail, the larger the number of matched terms between modified equations for initialisation and bulk methods, the smoother the obtained numerical solution. This is particularly manifest for numerical dissipation. Starting from the constraints to achieve time smoothness, which can quickly become prohibitive for they have to take the parasitic modes into consideration, we explain how the distinct lack of observability for certain lattice Boltzmann schemes -- seen as dynamical systems on a commutative ring -- can yield rather simple conditions and be easily studied as far as their initialisation is concerned. This comes from the reduced number of initialisation schemes at the fully discrete level. These theoretical results are successfully assessed on several lattice Boltzmann methods.
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