  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.
