Lattice Boltzmann open boundaries for hydrodynamic models

The mixed initial–boundary value problem for the equations of nonhydrostatic fluid dynamics is mathematically well posed, but is physically and computationally ill posed. The ill-posedness is exacerbated by boundary data taken from integrations of hydrostatic flow equations. The difficulty is found both in fully compressible flow and in Boussinesq flow; the latter is emphasized here. The mixed problem for the Boltzmann equation is well posed in every respect. Inserting lattice Boltzmann kinetics between hydrostatic boundary data and nonhydrostatic dynamics resolves the physical and computational ill-posedness of the dynamics, by relaxing the boundary data to nonhydrostatic dynamical consistency. Simple numerical experiments with boundary data taken from exact and approximate analytical solutions demonstrate the effectiveness of the approach.