Does the multiresolution lattice Boltzmann method allow to deal with waves passing through mesh jumps?

Abstract

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.

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

“…However, when the mesh is fixed in advance and a level jump is present, the solution obtained with lattice Boltzmann methods is known to experience artificial reflections and spurious phenomena; this is known to be a challenging problem. In fact, the formalism proposed in the present contribution allows to tackle this issue as well but requires a somewhat different setting which we have proposed in another short contribution [2]. We can show that the reflected part of the wave is of small amplitude and can be controlled, as it is proportional to ∆x 4 , when selecting γ = 1.…”