Solution of navier-stokes equations by goal programming

Abstract

Goal programming, a mathematical tool for the analysis of problems involving multiple, conflicting objectives arising in the fields of operations research and systems analysis, is employed to find the numerical solution of certain Navier-Stokes equations. As in the collocation method, the proposed technique involves approximating the unknown solution by a set of trial functions containing unknown coefficients. The technique then minimizes in a weighted residual sense the absolute value deviations of the differential equation residual by the modified pattern search algorithm for nonlinear goal programs. One important feature of this method in solving nonlinear problems is that it does not require the initial programming effort needed to set up a Newton method (or a similar approach) based upon the collocation approximation for the differential equation. Further, this approach is fundamentally more general than the collocation method because the number of undetermined parameters can be less than the number of spatial points. It is shown that the approximate solution to a Navier-Stokes equation with only a low order trial function compares favourably to other methods of weighted residual results.