Parametric schemes for the simulation of the advection process in finite-difference-based single-relaxation-time lattice Boltzmann methods

Abstract

The paper is devoted to the analysis of two parametric explicit finite-difference schemes for the linear advection equations, considered on the advection step of the splitting algorithm of the finite-difference-based single-relaxation-time lattice Boltzmann method. The schemes are constructed by the approximation of the terms with the spatial derivatives at the characteristic directions. It is demonstrated that by the proper choice of the parameter values, the third and fourth accuracy orders are realized.The stability analysis is based on the von Neumann method. As a result, the stability conditions as the inequalities on the values of the Courant–Friedrichs–Lewi number are obtained. It is demonstrated that the proposed schemes have better stability properties than the other high-order schemes and schemes with the spatial approximations at the Cartesian axes directions. It is demonstrated that the spurious numerical effects can be diminished by the proper choice of the parameter values.The obtained theoretical results are confirmed by the solution of numerical examples with the smooth and discontinuous initial conditions.