  We consider an adaptive multiresolution-based lattice Boltzmann scheme, which
we have recently introduced and studied from the perspective of the error
control and the theory of the equivalent equations. This numerical strategy
leads to high compression rates, error control and its high accuracy has been
explained on uniform and dynamically adaptive grids. However, one key issue
with non-uniform meshes within the framework of lattice Boltzmann schemes is to
properly handle acoustic waves passing through a level jump of the grid. It
usually yields spurious effects, in particular reflected waves. In this paper,
we propose a simple mono-dimensional test-case for the linear wave equation
with a fixed adapted mesh characterized by a potentially large level jump. We
investigate this configuration with our original strategy and prove that we can
handle and control the amplitude of the reflected wave, which is of fourth
order in the space step of the finest mesh. Numerical illustrations show that
the proposed strategy outperforms the existing methods in the literature and
allow to assess the ability of the method to handle the mesh jump properly.
