  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.
