Multiple-relaxation-times lattice Boltzmann models for nonlinear advection equation

Abstract

In this paper, we propose two multiple-relaxation-time lattice Boltzmann models for the nonlinear advection equation, which can be transformed into an advection-diffusion equation by introducing an anisotropic diffusion term. Through rigorous Chapman-Enskog analysis, we verify that the governing equation is correctly recovered from the present models. Comprehensive numerical simulations are performed across one-, two-, and three-dimensional scenarios, including linear/nonlinear advection equations and solid-phase transport problems, covering both smooth and discontinuous initial conditions. Accuracy order tests confirm that both proposed models exhibit second-order spatial accuracy. The numerical results demonstrate that the present models not only achieve excellent agreement with analytical solutions and existing numerical results for smooth cases but also outperform existing lattice Boltzmann models in handling discontinuous initial conditions, with significantly reduced numerical oscillations.