STABLE EXPLICIT STEPWISE MARCHING SCHEME IN ILL-POSED TIME-REVERSED 2D BURGERS' EQUATION.

Area of Science:

• Computational Fluid Dynamics
• Numerical Analysis
• Image Processing

Background:

• The 2D Burgers' equation is a fundamental model in fluid dynamics, often used to study wave propagation and shock formation.
• Solving the time-reversed version of this equation presents significant challenges due to its ill-posed nature, particularly when dealing with noisy or incomplete data.
• Accurate reconstruction of initial conditions from time-reversed evolution is crucial for various inverse problems, including image deblurring and signal recovery.

Purpose of the Study:

• To develop and analyze an unconditionally stable explicit numerical scheme for solving time-reversed 2D Burgers' initial value problems.
• To investigate the effectiveness of a compensating smoothing operator in stabilizing the explicit scheme.
• To demonstrate the scheme's capability in reconstructing initial conditions for a class of nonlinear problems, specifically for image deblurring applications.

Main Methods:

• Construction of an explicit difference scheme with backward time marching.
• Application of a compensating smoothing operator, based on (-Δ)^p with p > 2, to ensure numerical stability.
• Utilizing Fast Fourier Transform (FFT) algorithms for efficient synthesis of smoothing operators, potentially applicable to non-rectangular domains.
• Extensive numerical experiments, including deblurring of fictitiously blurred images generated from the forward 2D Burgers' equation.

Main Results:

• An unconditionally stable explicit difference scheme was successfully constructed for a specific class of time-reversed 2D Burgers' problems.
• The compensating smoothing operator effectively quenched instabilities, although it introduced some distortion from the true solution.
• Cumulative errors were found to be sufficiently small in many cases, allowing for useful and accurate results.
• The scheme demonstrated successful image deblurring capabilities, even with highly irregular intensity data and at high Reynolds numbers.

Conclusions:

• The developed explicit scheme offers a viable approach for solving certain time-reversed 2D Burgers' initial value problems.
• The stabilizing technique using smoothing operators is effective and adaptable to other ill-posed evolution equations.
• Numerical stability analysis based on linear problems proved applicable to the nonlinear 2D Burgers' equation.
• The method shows promise for practical applications like image deblurring, even with severely distorted data.