Stabilized leapfrog scheme run backward in time, and the explicit O(Δ t)² stepwise computation of ill-posed time-reversed 2D Navier–Stokes equations

Richardson's leapfrog scheme is notoriously unconditionally unstable in well-posed, forward, linear dissipative evolution equations. Remarkably, that scheme can be stabilized, marched backward in time, and provide useful reconstructions in 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. Eventually, this leads to a distortion away from the true solution. This is the stabilization penalty. In many interesting cases, that penalty is sufficiently small to allow for useful results. Effective smoothing operators based on (−Δ)p, with real p>2, can be efficiently synthesized using FFT algorithms. Similar stabilizing techniques were successfully applied in several other ill-posed evolution equations. The analysis of numerical stability is restricted to a related linear problem. However, as is found in leapfrog computations of well-posed meteorological and oceanic wave propagation problems, such linear stability is necessary but not sufficient in the presence of nonlinearities. Here, likewise, additional Robert–Asselin–Williams (RAW) time-domain filtering must be used to prevent characteristic leapfrog nonlinear instability unrelated to ill-posedness. Several 2D Navier–Stokes backward 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 significantly exceeding expectations.