Compensating operators and stable backward in time marching in nonlinear parabolic equations

Abstract

Linear and nonlinear parabolic advection–dispersion–reaction equations play a significant role in the geophysical sciences, where they are widely used in modeling contaminant transport in subsurface acquifers and rivers, as well as the dispersion of air pollutants from elevated sources. Environmental forensics aims at reconstructing the contaminant plume history, by solving the governing parabolic equations backward in time. However, such backward parabolic problems are exponentially ill-posed, and present serious computational difficulties. Step by step time-marching schemes are fundamental tools in the numerical exploration of well-posed nonlinear evolutionary partial differential equations. When the initial value problem is ill-posed however, such stepwise numerical schemes are necessarily unconditionally unstable and result in explosive noise amplification. This paper outlines a novel stabilized time-marching procedure for computing nonlinear parabolic equations on 2D rectangular regions, backward in time. Very little is known either analytically, or computationally, about this class of problems. To quench the instability, the procedure uses easily synthesized FFT-based compensating operators at every time step. A fictitious nonlinear image deblurring problem is used to evaluate the effectiveness of this computational approach. The method is compared with a previously introduced, global in time nonlinear Van Cittert iterative procedure. The latter is significantly more time consuming, and impractical on large problems.