Finite Difference Schemes for Incompressible Flows in the Velocity Impulse Density Formulation

together with perspectives on the application of real space renormalization procedures to vortex methods based on We consider finite difference schemes based on the impulse density variable. We show that the original velocity—impulse density the impulse density. formulation of Oseledets is marginally ill-posed for the inviscid flow, At a first sight, the impulse density formulation also proand this has the consequence that some ordinarily stable numerical videsanattractivewayof dealingwiththe issuesofboundary methods in other formulations become unstable in the velocity— conditions for numerical methods in the Eulerian frame, impulse density formulation. We present numerical evidence of such as finite difference and spectral methods. Methods this instability. We then discuss the construction of stable finite difference schemes by requiring that at the numerical level the based on vorticity formulations have to face the issue of ennonlinear terms be convertible to similar terms in the primitive forcing divergence-free conditions for velocity, vorticity, or variable formulation. Finally we give a simplified velocity—impulse the vector potential. Although in many cases easy solutions density formulation which is free of these complications and yet can be found (see for example [10]), this becomes a severe retains the nice features of the original velocity—impulse density limitation on the flexibility of the vorticity formulation. The formulation with regard to the treatment of boundary. We present numerical results on this simplified formulation for the driven velocity-pressure formulation, on the other hand, works cavity flow on both the staggered and non-staggered grids. Q 1997 well on staggered grid. However, boundary condition is still Academic Press an issue on non-staggered grids, particularly so when higher than second order methods are sought. The velocity—impulse density formulation seems to