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.

翻译：初始化方案的选择是一步扩展状态空间方法和多步方法的共同特征。本文聚焦于格子玻尔兹曼格式，该类格式可视为上述两类数值方案的具体实例。我们针对格子玻尔兹曼方法的初始化方案（由初始数据选取决定）提出修正方程分析。这些修正方程提供了指导准则，用于设计和分析初始化方案在目标柯西问题的一致性阶数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项数越多，所得数值解越光滑，这一现象在数值耗散方面尤为显著。从实现时间光滑性的约束条件出发（此类约束因需考虑寄生模式而可能迅速变得过高），我们阐释了为何特定格子玻尔兹曼格式（视为交换环上的动力系统）在可观性方面的显著缺失能够导出相当简化的条件，并使得其初始化问题易于研究——这得益于全离散层面初始化方案数量的减少。上述理论结果已在多种格子玻尔兹曼方法中得到成功验证。