Stable Explicit Time Marching in Well-Posed or Ill-Posed Nonlinear Parabolic Equations

This paper analyzes an effective technique for stabilizing pure explicit time dif­ ferencing in the numerical computation of multidimensional nonlinear parabolic equations. The method uses easily synthesized linear smoothing operators at each time step to quench the instabil­ ity. 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 ex­ plicit scheme requires no Courant restriction on the time step Δt, 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 rel­ atively 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 selfadjoint 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 modeled by advection dispersion equations. The paper uses fictitious mathematically blurred 512 × 512 pixel images as illustrative examples. Such images are associated with highly irregular jagged intensity data surfaces that can severely chal­ lenge ill-posed nonlinear reconstruction procedures. Instructive computational experiments demon­ strate the capabilities of the method in 2D rectangular regions.