Computational Design for Long-Term Numerical Integration of the Equations of Fluid Motion: Two-Dimensional Incompressible Flow. Part I

­z ­t 5 J(z, c), (2) The integral constraints on quadratic quantities of physical importance, such as conservation of mean kinetic energy and mean square vorticity, will not be maintained in finite difference analogues of or the equation of motion for two-dimensional incompressible flow, unless the finite difference Jacobian expression for the advection term is restricted to a form which properly represents the interaction ­ = 2c ­t 5 J(=2c, c), (3) between grid points, as derived in this paper. It is shown that the derived form of the finite difference Jacobian prevents nonlinear computational instability and thereby permits long-term numerical where J is the Jacobian operator with respect to the rectanintegrations.  1966 Academic Press gular coordinates, x and y, in the plane. When the Jacobian in this equation is replaced by spacedifferences of the usual form, INTRODUCTION