Explicit off-line criteria for stable accurate time filtering of strongly unstable spatially extended systems

Many contemporary problems in science involve making predictions based on partial observation of extremely complicated spatially extended systems with many degrees of freedom and physical instabilities on both large and small scales. Various new ensemble filtering strategies have been developed recently for these applications, and new mathematical issues arise. Here, explicit off-line test criteria for stable accurate discrete filtering are developed for use in the above context and mimic the classical stability analysis for finite difference schemes. First, constant coefficient partial differential equations, which are randomly forced and damped to mimic mesh scale energy spectra in the above problems are developed as off-line filtering test problems. Then mathematical analysis is used to show that under natural suitable hypothesis the time filtering algorithms for general finite difference discrete approximations to an sxs partial differential equation system with suitable observations decompose into much simpler independent s-dimensional filtering problems for each spatial wave number separately; in other test problems, such block diagonal models rigorously provide upper and lower bounds on the filtering algorithm. In this fashion, elementary off-line filtering criteria can be developed for complex spatially extended systems. The theory is illustrated for time filters by using both unstable and implicit difference scheme approximations to the stochastically forced heat equation where the combined effects of filter stability and model error are analyzed through the simpler off-line criteria.