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Stable explicit time marching in well-posed or ill-posed nonlinear parabolic equations.



Alfred S. Carasso


This paper analyzes an effective technique for stabilizing pure explicit time differencing in the numerical computation of multidimensional nonlinear parabolic equations. The method uses easily synthesized linear smoothing operators at each time step to quench the instability. Smoothing operators based on positive real powers of the negative Laplacian are helpful, and $(-\Δ)^p$ can be realized efficiently in rectangular domains using FFT algorithms. The stabilized explicit scheme requires no Courant restriction on the time step $\Δ t$, and is of great value in computing well-posed parabolic equations on fine meshes, by simply lagging the nonlinearity at the previous time step. Such stabilization leads to a distortion away from the true solution, but that error is often small enough to allow useful results in many problems of interest. The stabilized explicit scheme is also stable when run {\em backward in time}. This leads to relatively easy and useful computation of a significant, though limited, class of multidimensional nonlinear backward parabolic equations. Such backward reconstructions are of increasing interest in {\em environmental forensics}, where contaminant transport is often modeled by advection dispersion equations. In the canonical case of linear autonomous selfadjoint backward parabolic equations, with solutions satisfying prescribed bounds, it is proved that the stabilized explicit scheme can produce results that are nearly {\em best-possible}. The paper uses fictitious mathematically blurred $512 \times 512$ pixel images as illustrative examples. Such images are associated with highly irregular jagged intensity data surfaces that can severely challenge ill-posed nonlinear reconstruction procedures. Instructive computational experiments demonstrate the capabilities of the method in 2D rectangular regions.
NIST Interagency/Internal Report (NISTIR) - 8027
Report Number


FFT Laplacian stabilization, forward or backward nonlinear parabolic equations, non-integer power Laplacian, nonlinear image deblurring, quasi-reversibility method, stabilized explicit scheme, Van Cittert iteration.


Carasso, A. (2015), Stable explicit time marching in well-posed or ill-posed nonlinear parabolic equations., NIST Interagency/Internal Report (NISTIR), National Institute of Standards and Technology, Gaithersburg, MD, [online], (Accessed April 23, 2024)
Created July 27, 2015, Updated June 2, 2021