Data assimilation in 2D nonlinear coupled sound and heat flow, using a stabilized explicit finite difference scheme marched backward in time

TL;DR

This paper develops a stabilized explicit finite difference scheme for data assimilation in 2D coupled sound and heat flow systems, addressing ill-posedness and demonstrating its application to complex image data.

Contribution

It introduces a novel stabilized explicit finite difference method for backward time marching in nonlinear coupled systems, with analysis and practical examples.

Findings

Scheme effectively stabilizes backward time integration.

Successful reconstruction of irregular image data demonstrated.

Method applicable to general nonlinear problems.

Abstract

This paper considers the ill-posed data assimilation problem associated with hyperbolic/parabolic systems describing 2D coupled sound and heat flow. Given hypothetical data at time T > 0, that may not correspond to an actual solution of the dissipative system at time T, initial data at time t = 0 are sought that can evolve, through the dissipative system, into a useful approximation to the desired data at time T. That may not always be possible. A stabilized explicit finite difference scheme, marching backward in time, is developed and applied to nonlinear examples in non rectangular regions. Stabilization is achieved by applying a compensating smoothing operator at each time step, to quench the instability. Analysis of convergence is restricted to the transparent case of linear, autonomous, selfadjoint spatial differential operators. However, the actual computational scheme can be applied to more general problems. Data assimilation is illustrated using 512x512 pixel images. Such images are associated with highly irregular non smooth intensity data that severely challenge ill-posed reconstruction procedures. Successful and unsuccessful examples are presented.