An Accurate Multi-level Finite Difference Scheme for 1D Diffusion Equations Derived from the Lattice Boltzmann Method

An accurate and unconditionally stable explicit finite difference scheme for 1D diffusion equations is derived from the lattice Boltzmann method with rest particles. The system of the lattice Boltzmann equations for the distribution of the number of the fictitious particles is rewritten as a four-level explicit finite difference equation for the concentration of the diffused matter with two parameters. The consistency analysis of the four-level scheme shows that the two parameters which appear in the scheme, the relaxation parameter and the amount of rest particles, can be determined such that the scheme has the truncation error of fourth order. Numerical experiments demonstrate the fourth-order rate of convergence for various combinations of model parameters.