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

TL;DR

This paper demonstrates that an adaptive multiresolution lattice Boltzmann method can effectively manage wave transmission through mesh jumps, reducing spurious reflections and outperforming existing methods.

Contribution

The study introduces a simple test case and proves that their multiresolution lattice Boltzmann scheme controls reflected wave amplitudes at mesh jumps, with fourth-order accuracy.

Findings

The method reduces reflected wave amplitude to fourth order in the finest mesh space step.

Numerical results show superior performance compared to existing approaches.

The strategy effectively handles waves passing through mesh level 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.