COMPUTING ILL-POSED TIME-REVERSED 2D NAVIER-STOKES EQUATIONS, USING A STABILIZED EXPLICIT FINITE DIFFERENCE SCHEME...

This paper constructs an unconditionally stable explicit finite difference scheme, marching backward in time, that can solve an interesting but limited class of ill-posed, time-reversed, 2D incompressible Navier-Stokes initial value problems. Stability is achieved by applying a compensating smoothing operator at each time step to quench the instability. This leads to a distortion away from the true solution. However, in many interesting cases, the cumulative error is sufficiently small to allow for useful results. Effective smoothing operators based on (-∆) ^p , with real p > 2, can be efficiently synthesized using FFT algorithms. Similar stabilizing techniques were successfully applied in other ill-posed evolution equations. The analysis of numerical stabilty is restricted to a related linear problem. However, extensive numerical experiments indicate that such linear stability results remain valid when the explicit scheme is applied to a significant class of time-reversed nonlinear 2D Navier-Stokes initial value problems. Several reconstruction examples are included, based on the stream function-vorticity formulation, and focusing on 256 × 256 pixel images of recognizable objects. Such images, associated with non-smooth underlying intensity data, are used to create severely distorted data at time T > 0. Successful backward recovery is shown to be possible at parameter values exceeding expectations.