Lattice-Boltzmann methods are known for their simplicity, efficiency and ease of parallelization, usually relying on uniform Cartesian meshes with a strong bond between spatial and temporal discretization. This fact complicates the crucial issue of reducing the computational cost and the memory impact by automatically coarsening the grid where a fine mesh is unnecessary, still ensuring the overall quality of the numerical solution through error control. This work provides a possible answer to this interesting question, by connecting, for the first time, the field of lattice-Boltzmann Methods (LBM) to the adaptive multiresolution (MR) approach based on wavelets. To this end, we employ a MR multi-scale transform to adapt the mesh as the solution evolves in time according to its local regularity. The collision phase is not affected due to its inherent local nature and because we do not modify the speed of the sound, contrarily to most of the LBM/Adaptive Mesh Refinement (AMR) strategies proposed in literature, thus preserving the original structure of any LBM scheme. Besides, an original use of the MR allows the scheme to resolve the proper physics by efficiently controlling the accuracy of the transport phase. We carefully test our method to conclude on its adaptability to a wide family of existing lattice Boltzmann schemes, treating both hyperbolic and parabolic systems of equation, thus being less problem-dependent than the AMR approaches, which have a hard time granting an effective control on the error. The ability of the method to yield a very efficient compression rate and thus a computational cost reduction for solutions involving localized structures with loss of regularity is also shown, while guaranteeing a precise control on the approximation error introduced by the spatial adaptation of the mesh. The numerical strategy is implemented on a specific open-source platform called SAMURAI with a dedicated data-structure relying on set algebra.

中文翻译：

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基于多分辨率分析的带误差控制的多维全自适应格子玻尔兹曼方法

Lattice-Boltzmann方法以其简单，高效和易于并行化而著称，通常依赖于统一的笛卡尔网格，在空间和时间离散化之间具有很强的联系。这一事实使通过在不需要精细网格的情况下自动对网格进行粗化来降低计算成本和内存影响的关键问题变得复杂，仍然通过误差控制确保了数值解的整体质量。这项工作首次将晶格-玻尔兹曼方法（LBM）领域与基于小波的自适应多分辨率（MR）方法联系起来，为这个有趣的问题提供了可能的答案。为此，当解决方案根据其局部规律随时间变化时，我们采用MR多尺度变换来适应网格。由于其固有的局部性质，并且由于我们不修改声音的速度，所以与碰撞阶段无关，这与文献中提出的大多数LBM /自适应网格细化（AMR）策略相反，因此保留了任何LBM的原始结构方案。此外，MR的最初使用允许该方案通过有效地控制传输阶段的精度来解决适当的物理问题。我们仔细测试了我们的方法，以总结其对现有的大范围格子Boltzmann方案的适应性，同时处理双曲和抛物线方程组，因此，与AMR方法相比，其对问题的依赖性较小，因为AMR方法难以有效地控制错误。还显示了该方法产生非常有效的压缩率的能力，从而降低了涉及具有规则性损失的局部结构的解决方案的计算成本降低的能力，同时保证了对由网格空间适应引入的近似误差的精确控制。数值策略是在称为SAMURAI的特定开源平台上实施的，该平台具有依赖于集合代数的专用数据结构。