STABILIZED LEAPFROG SCHEME RUN BACKWARD IN TIME,AND THE EXPLICIT O(?t)^2 STEPWISE COMPUTATION OFILL-POSED TIME-REVERSED 2D NAVIER-STOKES EQUATIONS

Published

September 11, 2021

Author(s)

Alfred Carasso

Abstract

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 ap- plying 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 (−∆)^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 stabilty 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, addi- tional Robert-Asselin-Williams (RAW) time-domain filtering must be used to prevent characteristic leapfrog nonlinear instabilty, 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 at time T > 0. Successful backward recovery is shown to be possible at parameter values significantly exceeding expectations.

Citation

Inverse Problems in Science and Engineering

Pub Type

Journals

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Keywords

2D Navier-Stokes equations backward in time, High Reynolds numbers, Stabilizedbackward marching O(?t)^2 explicit scheme, Numerical experiments.

Citation

Carasso, A. (2021), STABILIZED LEAPFROG SCHEME RUN BACKWARD IN TIME,AND THE EXPLICIT O(?t)^2 STEPWISE COMPUTATION OFILL-POSED TIME-REVERSED 2D NAVIER-STOKES EQUATIONS, Inverse Problems in Science and Engineering, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=931597 (Accessed October 14, 2025)