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。本文聚焦于可被解释为上述两类数值格式实例的格子玻尔兹曼格式。针对由初始数据选择所确定的格子玻尔兹曼方法初始化方案，我们提出了一种修正方程分析。这些修正方程为基于与目标柯西问题的一致性阶次及数值解的时间光滑性来设计与分析初始化方案提供了指导准则。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——这一特征在数值耗散中尤为显著。从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难以处理），我们阐释了如何通过某些格子玻尔兹曼格式（视为交换环上的动力系统）所特有的可观测性显著缺失，来导出相当简明的条件，并对其初始化问题展开便捷研究。这一结果源于完全离散层面初始化方案数量的缩减。我们已在多种格子玻尔兹曼方法上成功验证了这些理论成果。