  A unified framework to derive discrete time-marching schemes for coupling of
immersed solid and elastic objects to the lattice Boltzmann method is
presented. Based on operator splitting for the discrete Boltzmann equation,
second-order time-accurate schemes for the immersed boundary method, viscous
force coupling and external boundary force are derived. Furthermore, a modified
formulation of the external boundary force is introduced that leads to a more
accurate no-slip boundary condition. The derivation also reveals that the
coupling methods can be cast into a unified form, and that the immersed
boundary method can be interpreted as the limit of force coupling for vanishing
particle mass. In practice, the ratio between fluid and particle mass
determines the strength of the force transfer in the coupling. The integration
schemes formally improve the accuracy of first-order algorithms that are
commonly employed when coupling immersed objects to a lattice Boltzmann fluid.
It is anticipated that they will also lead to superior long-time stability in
simulations of complex fluids with multiple scales.
