Wavelet Based Spatial Scaling of Coupled Reaction Diffusion Fields

Multiscale schemes for transferring information from fine to coarse scales are typically based on some sort of averaging. Such schemes smooth the fine scale features of the underlying fields, thus altering the fine scale correlations. As a superior alternative to averaging, a wavelet based scheme for the exchange of information between a reactive and diffusive field in the context of multiscale reaction-diffusion problems is proposed and analyzed. The scheme is shown to be efficient in passing information along scales, from fine to coarse, i.e. up-scaling as well as from coarse to fine, i.e. down-scaling. In addition, it retains fine scale statistics, mainly due to the capability of wavelets to represent fields hierarchically. Critical to the success of the scheme is the identification of dominant scales containing the majority of useful information. The scheme is applied in detail to the analysis of a diffusive system with chemically reacting boundary. Reactions are simulated using kinetic Monte Carlo (KMC) and diffusion is solved by finite differences. Spatial scale differences are present at the interface of the KMC sites and the diffusion grid. The computational efficiency of the scheme is compared to results obtained by local averaging, and to results from a benchmark model. The spatial scaling scheme ties to wavelet based schemes for temporal scaling, presented elsewhere by the authors.

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