Upwind techniques via variational principles

Published in Int. J. Number. Meth. Engng. Vol. 28 (4), pp. 769-784, 1989
doi: 10.1002/nme.1620280404

Abstract

The amount of upwind or magnitude of off‐centering needed in the numerical solution of second order differential equations with significant first derivatives is justified more rigorously in this paper by requiring the satisfaction of a variational principle. It will be shown that, for a discrete solution, a set of variational principles can be found which allow the problem to be solved as a self‐adjoint system. The technique presented here will give a clear indication of those regions where (i) upwind techniques are not required, (ii) upwind techniques are necessary and sufficient and (iii) upwind techniques are insufficient and artificial viscosity is required.