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 numericalsolution. 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, 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 numberof initialisation schemes at the fully discrete level. These theoretical results are successfully assessed on several lattice Boltzmann methods.

翻译：初始化的选择自由度是单步扩展状态空间方法与多步方法共有的特征。本文聚焦于格子玻尔兹曼格式，这类格式可被视作前述两类数值方法的实例。我们提出了一种针对格子玻尔兹曼方法初始化格式的修正方程分析，该分析由初始数据的选择决定。这些修正方程提供了指导依据，用于设计和分析初始化相对于目标柯西问题的一致阶次以及数值解的时间光滑性。具体而言，初始化修正方程与体方法修正方程之间匹配的项数越多，所获得的数值解就越光滑，这在数值耗散中尤为明显。从实现时间光滑性的约束条件（这些条件可能迅速变得过于苛刻）出发，我们阐释了某些格子玻尔兹曼格式——被视为交换环上的动力系统——在可观性上的显著缺失如何能产生相当简单的条件，并使得其初始化问题易于研究。这源于在完全离散层面初始化格式数量的减少。这些理论结果在多个格子玻尔兹曼方法上得到了成功验证。