The book introduces the fundamentals and applications of the lattice Boltzmann method (LBM) for incompressible viscous flows. It is written clearly and easy to understand for graduate students and researchers.The book is organized as follows. In Chapter 1, the SRT- and MRT-LBM schemes are derived from the discrete Boltzmann equation for lattice gases and the relation between the LBM and the Navier-Stokes equation is explained by using the asymptotic expansion (not the Chapman-Enskog expansion). Chapter 2 presents the lattice kinetic scheme (LKS) which is an extension method of the LBM and can save memory because of needlessness for storing the velocity distribution functions. In addition, an improved LKS which can stably simulate high Reynolds number flows is presented. In Chapter 3, the LBM combined with the immersed boundary method (IB-LBM) is presented. The IB-LBM is well suitable for moving boundary flows. In Chapter 4, the two-phase LBM is explained from the point of view of the difficulty in computing two-phase flows with large density ratio. Then, a two-phase LBM for large density ratios is presented. In Appendix, sample codes (available for download) are given for users.Key Features• The relation between the LBM and the Navier-Stokes equation is derived by using the asymptotic expansion with a diffusive time scale (the authors’ original work). The Chapman–Enskog expansion is usually used for that derivation, but the ordering of the variables as well as the time scale is not clear. On the other hand, the asymptotic expansion is easy to understand and makes clear the ordering of the variables. The asymptotic expansion is used through the book• The LKS (also the authors’ original work) is an extension method of the LBM and needs no velocity distribution functions. Also, an improved LKS can stably simulate high Reynolds number flows• The IB-LBM (also authors’ original work) is useful for simulations of moving boundary flows. For numerical examples, free flight simulations of butterfly-like and dragonfly-like flapping wing-body models are illustrated• Two-Phase LBM for large density ratios (also authors’ original work) is presented and applied to the simulation of a milk crown• Sample codes (available for download) for the IB-LBM and the two-phase LBM are given in Appendix, and the codes are useful for users

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