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：对初始化方案选择的自由度是单步扩展状态空间方法与多步方法的共同特征。本文聚焦于格子玻尔兹曼格式——该类格式可被解读为前述两类数值格式的典型范例。我们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，提出了修正方程分析方法。这些修正方程为根据数值解相对于目标柯西问题的一致阶及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修正方程相匹配的项数越多，所获得的数值解就越光滑。这一特性在数值耗散现象中尤为显著。基于实现时间光滑性所需的约束条件——这些约束因需考虑寄生模式而可能迅速变得难以承受——我们阐释了为何某些格子玻尔兹曼格式（视为交换环上的动力系统）在初始化问题上能呈现出相当简洁的条件并易于研究，其根源在于完全离散层次上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。