Numerical methods for incompressible fluid dynamics have developed to the point at which a survey of the field is both timely and appropriate. A major stimulus to the field has been the large number of applications in which incompressible flows play a crucial role, and this has spurred the interest of numerous computational engineers and mathematicians. The articles which follow provide a reasonably broad view of algorithmic and theoretical aspects of incompressible flow calculations.

It should be noted at the outset that it can be dangerous to define an algorithm for simulating incompressible flows by setting, for example, the density to be constant in a successful compressible flow algorithm. The nature of the pressure as a Lagrange multiplier rather than as a thermodynamic variable as well as the infinite speed of propagation of disturbances and other factors peculiar to incompressible flows, make algorithmic development and implementation in this context a unique undertaking (see Appendix 7A).

Perhaps the first major advance in the application of large scale digital computation to incompressible flows occured in the late 1950s with the introduction of staggered mesh techniques, exemplified, for example, by the Marker-and-Cell (MAC) scheme. The use of staggered meshes in the context of the primitive variable formulation was found to provide a stable discretization of the incompressibility constraint. Shortly thereafter, it was realized that the use of staggered meshes could be avoided by employing the streamfunction-vorticity formulation in which the incompressibility constraint does not explicitly appear. Numerous finite difference algorithms were proposed and used based on this formulation of the Navier-Stokes equations.