Study on Localized Exponential Time Differencing Methods for Evolution Partial Differential Equations

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Many important physical phenomena are modeled by semilinear or fully nonlinear evolution partial differential equations. The overall goal of the project is to enhance the efficiency and scalability of exponential integrator-based methods for solving these equations by designing and analyzing highly scalable localized exponential time differencing methods and to apply them to numerically simulate and investigate a wide range of related application problems in science and engineering. The proposed work is of practical interest with significant influences as the developed methods are highly scalable on modern supercomputer systems, and can serve as an efficient, accurate and stable computational tool for simulations of these stiff problems. Direct and transformative innovations resulting from the project will greatly improve modeling and computational capabilities for many fields, such as design of new materials and oil recovery from fractured oil reservoirs. In addition, this project will also offer a unique educational opportunity for graduate students with interests in computational and applied mathematics by having them participate in an interdisciplinary research environment.Direct parallelization of global exponential time differencing methods is often very hard to be scalable on massively distributed systems due to the intensive data communications needed by fast Fourier transform or by Krylov subspace-based calculations for products of matrix exponentials and vectors. On the other hand, domain decomposition approaches have been well established for many classic time integration methods, but not enough attention and work have been devoted to exponential integrators. This project involves a thorough study on the development and analysis of iterative and noniterative localized exponential time differencing methods based on domain decomposition, with a family of time-dependent scalar diffusion equations as the prototype problem. The PI will also apply the developed methods to study some phase field models for multi-component and multi-phase systems arising from materials science and petroleum engineering. This project would offer new insights through numerical investigations to the understanding of the macroscopic properties and reliability of alloys and the physical phenomena (such as liquid droplets, gas bubbles, and capillary pressure) of hydrocarbon fluids.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.