Problems of potential flow in irregular domains bounded by rigid solids and satisfying perhaps conditions at infinity require numerical methods. Computers and software are now so powerful that it can be easier to compute a solution than to find the exact one in a reference book. There are many techniques that may be used to solve Laplace's equation with prescribed boundary conditions. These techniques are readily available even in “search” on the web.

The numerical simulation of the deformation of interfaces between two immiscible fluids or in gas–liquid flows is currently an active topic of research and many options are available for researchers. Level-set methods associated with the names of S. Osher, R. Fedkiw, and J. Sethian, volume-of-fluid methods associated with the name of S. Zaleski, and front-tracking methods associated with the name of G. Trygvasson, are high among the most popular methods. Readers can find references in the comprehensive reviews by Yeung (1982), Tsai and Yue (1996), and Scardovelli and Zaleski (1999), or in “search” on Google.

Perturbation methods

The problem of numerical simulation of the shape of free surfaces in potential flows of inviscid fluids has been considered by various authors. Perturbation methods for nonlinear irrotational waves on an inviscid fluid were introduced by Stokes (1847). He expanded the solution in powers of the amplitude.