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：在单步扩展状态空间方法和多步方法中，初始化方案的选择具有共同的自由度。本文聚焦于格子玻尔兹曼格式——这类格式可被视作上述两类数值格式的典型实例。我们针对格子玻尔兹曼方法中由初始数据选择决定的初始化方案，提出了修正方程分析方法。这些修正方程为设计并分析初始化方案提供准则，涉及与目标柯西问题的一致性阶数以及数值解的时间光滑性。具体而言，初始化修正方程与体修正方程匹配项数越多，所得数值解越光滑，这一现象在数值耗散中尤为显著。从实现时间光滑性的约束条件出发（这类条件因需考虑寄生模态而可能迅速变得难以满足），我们阐释了为何某些格子玻尔兹曼格式——被视为交换环上的动力系统——在可观测性上的特定缺失，能够产生相当简明的初始化条件并易于分析。这源于全离散层面初始化方案数量的减少。这些理论结果在多种格子玻尔兹曼方法上得到了成功验证。