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Lattice Boltzmann Methods (LBM) stand out for their simplicity andcomputational efficiency while offering the possibility of simulating complexphenomena. While they are optimal for Cartesian meshes, adapted meshes havetraditionally been a stumbling block since it is difficult to predict the rightphysics through various levels of meshes. In this work, we design a class offully adaptive LBM methods with dynamic mesh adaptation and error controlrelying on multiresolution analysis. This wavelet-based approach allows toadapt the mesh based on the regularity of the solution and leads to a veryefficient compression of the solution without loosing its quality and with thepreservation of the properties of the original LBM method on the finest grid.This yields a general approach for a large spectrum of schemes and allowsprecise error bounds, without the need for deep modifications on the referencescheme. An error analysis is proposed. For the purpose of assessing theapproach, we conduct a series of test-cases for various schemes and scalar andsystems of conservation laws, where solutions with shocks are to be found andlocal mesh adaptation is especially relevant. Theoretical estimates areretrieved while a reduced memory footprint is observed. It paves the way to animplementation in a multi-dimensional framework and high computationalefficiency of the method for both parabolic and hyperbolic equations, which isthe subject of a companion paper.

Multiresolution-based mesh adaptation and error control for lattice Boltzmann methods with applications to hyperbolic conservation laws