Lattice Boltzmann Methods (LBM) stand out for their simplicity and computational efficiency while offering the possibility of simulating complex phenomena. While they are optimal for Cartesian meshes, adapted meshes have traditionally been a stumbling block since it is difficult to predict the right physics through various levels of meshes. In this work, we design a class of fully adaptive LBM methods with dynamic mesh adaptation and error control relying on multiresolution analysis. This wavelet-based approach allows to adapt the mesh based on the regularity of the solution and leads to a very efficient compression of the solution without loosing its quality and with the preservation of the properties of the original LBM method on the finest grid. This yields a general approach for a large spectrum of schemes and allows precise error bounds, without the need for deep modifications on the reference scheme. An error analysis is proposed. For the purpose of assessing the approach, we conduct a series of test-cases for various schemes and scalar and systems of conservation laws, where solutions with shocks are to be found and local mesh adaptation is especially relevant. Theoretical estimates are retrieved while a reduced memory footprint is observed. It paves the way to an implementation in a multi-dimensional framework and high computational efficiency of the method for both parabolic and hyperbolic equations, which is the subject of a companion paper.

中文翻译：

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基于多分辨率的网格自适应和网格Boltzmann方法误差控制及其在双曲守恒律中的应用

格子Boltzmann方法（LBM）以其简单性和计算效率而著称，同时提供了模拟复杂现象的可能性。尽管它们是笛卡尔网格的最佳选择，但由于很难通过各种级别的网格来预测正确的物理学，适应性网格传统上一直是绊脚石。在这项工作中，我们基于多分辨率分析设计了一类具有动态网格自适应和误差控制的完全自适应LBM方法。这种基于小波的方法允许根据解决方案的规则性来调整网格，并导致解决方案非常有效的压缩，而不会降低其质量，并且在最细的网格上保留了原始LBM方法的属性。这产生了适用于各种方案的通用方法，并允许精确的误差范围，无需对参考方案进行深入修改。提出了错误分析。为了评估该方法，我们针对各种方案，标量和守恒定律系统进行了一系列测试用例，其中将找到具有冲击的解决方案，并且局部网格自适应尤其重要。检索理论估计值，同时观察到内存占用减少。它为在抛物线和双曲线方程组的多维框架中实现该方法和高计算效率铺平了道路，这是本文的主题。在哪里可以找到具有冲击的解决方案，而局部网格自适应尤其重要。检索理论估计值，同时观察到内存占用减少。它为在抛物线和双曲线方程组的多维框架中实现该方法和高计算效率铺平了道路，这是本文的主题。在哪里可以找到具有冲击的解决方案，而局部网格自适应尤其重要。检索理论估计值，同时观察到内存占用减少。它为在抛物线和双曲线方程组的多维框架中实现该方法和高计算效率铺平了道路，这是本文的主题。