Multiple solutions of finite difference approximations to some steady problems of fluid dynamics

It is pointed out that a nonlinear boundary value problem that models a steady-state physical problem will often be known to have a unique solution. Certain finite difference schemes, however, will possess multiple solutions when applied to the problem. It is reasonable to assume in this case that there is one finite difference solution that best approximates the solution of the boundary value problem and that the other solutions may be regarded as spurious. The existence of multiple finite difference solutions leads to computational problems because having completed a solution, it is not always clear whether or not it is spurious. Results obtained concerning multiple solutions of certain finite difference approximations to certain steady-state equations that model fluid problems are discussed. Particular attention is given to the uniqueness and nonuniqueness of finite difference solutions for Burgers equation, one-dimensional duct flow, and the incompressible Navier-Stokes equations with homogeneous boundary conditions.