Temporal Splitting Methods for Multiscale Problems

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In many physical systems of practical interest, phenomena occur in heterogeneous media with properties varying at multiple scales and having disparate values on each scale. Examples include multi-physics processes in filters, membranes, and Earth's subsurface. Standard numerical approaches for simulating these phenomena to obtain accurate predictions require tremendous computational effort. The goal of this project is to develop a unified framework for accurately and efficiently simulating complex multiscale, time dependent physical phenomena that involve flow, transport, and mechanical deformations that arise in porous media. The project will support education by training a new generation of computational mathematicians who work in multidisciplinary research.This project involves the development and analyses of novel temporal splitting methods that are designed to overcome challenges that arise when simulating multiscale, time-dependent physical phenomena. The methods are based on solution decomposition for non-stationary multiscale models and will consider space and time heterogeneities that are highly coupled. The goal is to provide a general framework that combines temporal splitting algorithms and spatial multiscale decompositions with a rigorous theoretical analysis of the new algorithms. The specific objectives of the project are: (i) to study temporal splitting algorithms for simulations of non-stationary multiscale models; (ii) to understand space and time interaction in multiscale models; (iii) to analyze temporal splitting approaches to guide the choice for space decomposition; (iv) to design novel splitting approaches for nonlinear models; and (v) to test and demonstrate proposed approaches for improving predictions of multiscale, time-dependent physical phenomena in engineering and geosciences.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.