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

Abstract

ABSTRACTThis paper constructs an unconditionally stable explicit difference scheme, marching backward in time, that can solve a limited, but important 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 , 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 stability 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 2D Burgers' initial value problems. As illustrative examples, the paper uses fictitiously blurred 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. The stabilized explicit scheme, applied to the time-reversed 2D Burgers' equation, is then used to deblur these images. Examples involving simpler data are also studied. Successful recovery from severely distorted data is shown to be possible, even at high Reynolds numbers.