High accuracy analysis of adaptive multiresolution-based lattice Boltzmann schemes via the equivalent equations

Multiresolution provides a fundamental tool based on the wavelet theory to build adaptive numerical schemes for Partial Differential Equations and time-adaptive meshes, allowing for error control. We have previously introduced this strategy to construct adaptive lattice Boltzmann methods with this interesting feature. Furthermore, these schemes allow for an effective memory compression of the solution when spatially localized phenomena (such as shocks or fronts) are involved, to rely on the original scheme without any manipulation at the finest level of grid and to reach a high level of accuracy on the solution. Nevertheless, the peculiar way of modeling the desired physical phenomena in the lattice Boltzmann schemes calls, besides the possibility of controlling the error introduced by the mesh adaptation, for a deeper and more precise understanding of how mesh adaptation alters the target equations describing the physics. In this contribution, this issue is studied by performing the equivalent equations analysis of the adaptive method after writing the scheme under an adapted formalism. It provides an essential tool to master the perturbations introduced by the adaptive numerical strategy, which can thus be devised to preserve the desired features of the reference scheme at a high order of accuracy. The theoretical considerations are corroborated by numerical experiments in both the 1D and 2D context, showing the relevance of the analysis. In particular, we show that our numerical method outperforms traditional approaches, whether or not the solution of the reference scheme converges to the solution of the target equation. Furthermore, we discuss the influence of various collision strategies for non-linear problems, showing that they have only a marginal impact on the quality of the solution, thus further assessing the proposed strategy.