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On the Stability of Multi-Step Finite-Difference-Based Lattice Boltzmann Schemes

Stability of finite-difference-based off-lattice Boltzmann schemes is analyzed. The time derivative in system of discrete Boltzmann equations is approximated by two-step modified central difference. Advective term is approximated by finite differences from first- to fourth-orders of accuracy. Characteristics-based (CB) schemes and schemes with traditional separate approximations of space derivatives are considered. A special class of high-order CB schemes with approximation in the internal nodes of grid patterns is constructed. It is demonstrated that apparent viscosity for the schemes of high-order is equal to kinematic viscosity of the system of Bhatnaghar–Gross–Krook kinetic equations. Stability of the schemes is analyzed by the von Neumann method for the cases of two flow regimes in unbounded domain. Stability is analyzed by the investigation of the stability domains in parameter space. The area of the domain is considered as the main numerical characteristic of the stability. As the main result of the analysis, it must be mentioned that the areas of CB schemes are greater than areas for the schemes with separate approximations.