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# Stable explicit stepwise marching scheme in ill-posed time-reversed 2D Burgers' Equation

Published

### Author(s)

Alfred S. Carasso

### Abstract

This paper constructs an unconditionally stable explicit difference scheme, marching backward in time, that can solve an important, but limited, class of time-reversed 2D Burgers' 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 $(-\Δ)^p$, with real $p > 2$, can be efficiently synthesized using FFT algorithms, and this may be feasible even in non-rectangular regions. Similar stabilizing techniques were successfully applied in other ill-posed evolution equations. The analysis of numerical stabilty is restricted to a related linear problem. Extensive numerical experiments indicate such linear stability results remain valid when the explicit scheme is applied to a significant class of time-reversed nonlinear 2D Burgers' initial value problems. As illustrative examples, the paper uses fictitiously blurred $256 \times 256$ pixel images, obtained by using sharp images as initial values in well-posed, forward 2D Burgers equations. Such images are associated with highly irregular underlying intensity data that can seriously challenge ill-posed reconstruction procedures. Deblurring these images proceeds by applying the stabilized explicit scheme on the corresponding time-reversed 2D Burgers' equation. Successful recovery from severely distorted data is shown to be possible, even at high Reynolds numbers.
Citation
Inverse Problems in Science and Engineering
Volume
27

### Keywords

2D viscous Burgers' equation backward in time, High Reynolds numbers, Stabilized explicit marching difference scheme, Numerical experiments.

## Citation

Carasso, A. (2018), Stable explicit stepwise marching scheme in ill-posed time-reversed 2D Burgers' Equation, Inverse Problems in Science and Engineering, [online], https://doi.org/10.1080/17415977.2018.1523905. (Accessed March 22, 2023)
Created August 31, 2018, Updated September 25, 2020