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

Abstract

This paper analyses 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 can be realized efficiently in rectangular domains using FFT algorithms. The stabilized explicit scheme requires no Courant restriction on the time step , 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. However, that error is often small enough to allow useful results in many problems of interest. The stabilized explicit scheme is also stable when run backward in time. This allows for relatively easy and useful computation of a significant class of multidimensional nonlinear backward parabolic equations, and complements the quasi-reversibility method. In the canonical case of linear autonomous self-adjoint backward parabolic equations, with solutions satisfying prescribed bounds, it is proved that the stabilized explicit scheme can produce results that are nearly best possible. Such backward reconstructions are of increasing interest in environmental forensics, where contaminant transport is often modelled by advection dispersion equations. The paper uses fictitious mathematically blurred 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.