  In this paper, a kind of finite-difference lattice Boltzmann method with the
second-order accuracy of time and space (T2S2-FDLBM) is proposed. In this
method, a new simplified two-stage fourth order time-accurate discretization
approach is applied to construct time marching scheme, and the spatial gradient
operator is discretized by a mixed difference scheme to maintain a second-order
accuracy both in time and space. It is shown that the previous
finite-difference lattice Boltzmann method (FDLBM) proposed by Guo [1] is a
special case of the T2S2-FDLBM. Through the von Neumann analysis, the stability
of the method is analyzed and two specific T2S2-FDLBMs are discussed. The two
T2S2-FDLBMs are applied to simulate some incompressible flows with the
non-uniform grids. Compared with the previous FDLBM and SLBM, the T2S2-FDLBM is
more accurate and more stable. The value of the Courant-Friedrichs-Lewy
condition number in our method can be up to 0.9, which also significantly
improves the computational efficiency.
