No abstract

In this paper, in order to expand the lattice Boltzmann method (LBM) to deal with more space-fractional systems, a fresh lattice Boltzmann scheme is proposed to approximate a Lévy–Feller advection–diffusion process, which is governed by the Lévy–Feller fractional advection–diffusion equation (LFADE). First, the fractional integral operator is discretized and the LFADE is transformed into a standard equation. Second, combining with Taylor expansion and Chapman–Enskog analysis, a family of the LFADE is recovered correctly from the continuous Boltzmann equation through selecting the equilibrium distribution functions. Finally, some test examples are presented and it is found that the numerical results agree well with the analytical solutions. In addition, the result in terms of stability is also tested by comparing with previous studies.

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A novel numerical scheme depending on Lattice Boltzmann method is proposed to solve the Sharma–Tasso–Olver equation in one, two and three dimension. The local equilibrium distribution functions and modified functions are obtained via Taylor expansion and Chapman–Enskog multiscale expansion techniques. The macro equation is recovered accurately based on the above functions, and the stability conditions of the equations are deduced. In addition, through simulating numerically some of the initial‐boundary value problems of multidimensional STO equations, the results demonstrate that the numerical solutions and the exact solutions tend to be identical. It also indicates that the model is feasible within a specific range and provides a reliable the approach for solving the multidimensional STO equations.