COMPUTING ILL-POSED TIME-REVERSED 2D NAVIER-STOKES EQUATIONS, USING A STABILIZED EXPLICIT DIFFERENCE SCHEME MARCHING BACKWARD IN TIME

Published

December 6, 2019

Author(s)

Alfred S. Carasso

Abstract

Abstract. This paper constructs an unconditionally stable explicit 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 2 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.

Citation

Inverse Problems in Science and Engineering

Pub Type

Journals

Download Paper

https://doi.org/10.1080/17415977.2019.1698564

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Keywords

2D Navier-Stokes equations backward in time, High Reynolds numbers, Stabilized explicit marching difference scheme, Numerical experiments.words. explicit marching difference scheme. Numerical experiments.

Citation

Carasso, A. (2019), COMPUTING ILL-POSED TIME-REVERSED 2D NAVIER-STOKES EQUATIONS, USING A STABILIZED EXPLICIT DIFFERENCE SCHEME MARCHING BACKWARD IN TIME, Inverse Problems in Science and Engineering, [online], https://doi.org/10.1080/17415977.2019.1698564 (Accessed December 4, 2024)