Numerical Integration of Navier-Stokes Equations

T is extensive literature on the numerical integration of some difference forms of the Navier-Stokes equations. The references quoted herein are those relevant to later discussions and not intended to be complete. There is also extensive literature on the mathematics of difference methods for the solution of partial differential equations for which Ref. 1 is a comprehensive review and contains an extensive bibliography. The present paper is an attempt to develop the implications of a mathematical theorem, which, when put into proper perspective, offers a unified and coherent treatment of various practical aspects of computation. Like approximate differential analyses and physical tests, numerical methods are fallible. Larger and faster computers offer no easy answer to computational difficulties like stability and convergence. Smooth and physically reasonable results of computation are often less accurate than those not so smooth. Since the true asymptotic nature is difficult to establish, both the difference and the differential approximations are nonrigorous; but they are useful, especially with the help of physical experimentation and rigorous mathematical results. We are sympathetic to such heuristic and nonrigorous analysis in favor of obtaining results useful in practice. We shall consider only the difference form of the NavierStokes equations in Eulerian coordinates. Lagrangian coordinates are convenient for flows involving free-surface boundary or active processes associated with fluid elements. It suffers, however, from the serious distortions of the Lagrangian net and from the cummulative errors of the particle paths at large times. Thus various mixed or coupled Eulerian-Lagrangian schemes have been developed even for free boundary problems. Eulerian formulation, even for a single fluid in the absence of a free boundary, has its share of problems, which we shall discuss. We write, for the NavierStokes equations,