A lattice-adaptive model for solving acoustic wave equations based on the lattice Boltzmann method

Abstract

This paper addresses the need for an efficient and adaptable approach to solving linear acoustic wave equations in the lattice Boltzmann method (LBM). A novel lattice-adaptive model is introduced, derived through a Chapman–Enskog analysis, which utilizes a single relationship for the equilibrium distribution function across all lattice structures. The intended derivation begins by considering a standard equilibrium distribution function with unknown coefficients. By selecting the displacement of the acoustic wave as the zero-order microscopic moment, accurate recovery of the macroscopic wave equation is ensured. Unlike existing methods, the model simplifies the complexity associated with equilibrium distribution functions and offers greater versatility. The model is validated through extensive benchmark testing on one and two-dimensional wave propagation problems. Results demonstrate excellent agreement with analytical solutions, with maximum root mean square errors of 10-3 (<0.1% error) and minimum errors of 10-6 (<0.0001% error), indicating high predictive accuracy (>99.9%). Additionally, the model exhibits second-order spatial accuracy, with the relative error norm E_2 displaying slopes close to 2, signifying a spatial accuracy of second order. The numerical simulations show a decrease in errors as the mesh size becomes more refined.