Data assimilation in 2D viscous Burgers equation using a stabilized explicit finite difference scheme run backward in time

Abstract

The 2D viscous Burgers equation is a system of two nonlinear equations in two unknowns, . This paper considers the data assimilation problem of finding initial values that can evolve into a close approximation to a desired target result , at some realistic T>0. Highly nonsmooth target data are considered, that may not correspond to actual solutions at time T. Such an ill-posed 2D viscous Burgers problem has not previously been studied. An effective approach is discussed and demonstrated based on recently developed stabilized explicit finite difference schemes that can be run backward in time. Successful data assimilation experiments are presented involving 8 bit, pixel grey-scale images, defined by nondifferentiable intensity data. An instructive example of failure is also included.