Single-step lattice Boltzmann scheme families dependent on a parameter are proposed. The construction of the schemes is based on the integral form of the system of kinetic equations obtained by the method of discrete velocities. The schemes are developed using the quadrature formulas of zero and first algebraic degrees. In several cases of parameter values, the second order of approximation takes place. The Lyapunov stability study of the proposed schemes is performed for the case of the absence of coordinate dependency. Two model hydrodynamic problems are considered to illustrate the application of the second-order implicit scheme.

Keywords: kinetic equations, lattice Boltzmann schemes, single-step methods

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