COMPUTING ILL-POSED TIME-REVERSED 2D NAVIER-STOKES EQUATIONS, USING A STABILIZED EXPLICIT FINITE DIFFERENCE SCHEME
1Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, MD 20899.
Inverse Problems in Science and Engineering
|June 16, 2021
Summary
This study presents a stable explicit finite difference scheme for solving time-reversed 2D Navier-Stokes equations. Despite introducing smoothing, it yields useful results for ill-posed problems, enabling image reconstruction.
Area of Science:
- Computational Fluid Dynamics
- Numerical Analysis
- Image Reconstruction
Background:
- Ill-posed problems in fluid dynamics, particularly time-reversed 2D incompressible Navier-Stokes equations, pose significant challenges.
- Existing numerical methods often struggle with stability and accuracy when dealing with such time-reversed scenarios.
Purpose of the Study:
- To develop an unconditionally stable explicit finite difference scheme for solving a class of ill-posed, time-reversed 2D Navier-Stokes initial value problems.
- To enable the reconstruction of useful data from distorted, time-reversed fluid dynamics scenarios.
Main Methods:
- Construction of an explicit finite difference scheme with backward time marching.
- Application of a compensating smoothing operator, based on (-∆)^p with p > 2, at each time step to ensure stability.
- Efficient synthesis of smoothing operators using Fast Fourier Transform (FFT) algorithms.
- Numerical stability analysis restricted to a related linear problem, validated by extensive experiments on nonlinear problems.
Main Results:
- The developed scheme achieves unconditional stability for the targeted class of problems.
- While the smoothing introduces distortion, cumulative errors remain small enough for useful results in many cases.
- The method, using the stream function-vorticity formulation, successfully reconstructs recognizable objects from severely distorted 256x256 pixel images.
- Backward recovery is demonstrated to be possible at parameter values exceeding initial expectations.
Conclusions:
- The explicit finite difference scheme offers a viable approach for tackling ill-posed, time-reversed 2D Navier-Stokes problems.
- The stabilizing technique, though introducing minor distortions, allows for practical and useful reconstructions.
- The method shows promise for applications in image reconstruction and solving other ill-posed evolution equations.
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