Shape-Morphing Modes for Efficient Computation of Multiscale Evolution Partial Differential Equations with Conserved Quantities

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Large-scale computations are needed in many areas of science and engineering, such as climate modeling, weather forecast, and design of sustainable structures. The corresponding mathematical models often involve a wide range of time and spatial scales which the simulations need to resolve. Efficiently resolving these multiscale structures has been a long-standing challenge in scientific computing. This project proposes shape-morphing modes as a new computational method that will drastically reduce the computational time and memory requirements of simulating multiscale systems. Shape-morphing modes are computational elements that adaptively change their shape and location to efficiently capture various temporal and spatial scales. The resulting computational speedup will enable us to perform real-time prediction, optimization, and control tasks that had been inaccessible to previous methods. The dynamics of spatiotemporal systems are routinely described by time-dependent partial differential equations (PDEs). The solutions of these PDEs often exhibit time-varying localized structures, with sharp gradients, surrounded by regions of large-scale motion. Such multiscale PDEs arise in numerous applications, such as aircraft design, weather prediction, ocean and climate modeling, where resolving small scale structures remains a major challenge. Currently, there are two broad classes of methods for addressing this challenge: 1. Adaptive methods which dynamically evolve the spatial discretization so that the computational grid is refined around the localized structure and less so in the quiescent regions. 2. Multiresolution methods, such as wavelets, which encode various scales in the basis instead of the discretization. This project will develop a new and computationally efficient method called shape-morphing modes. The main idea behind this method is to use a time-dependent basis of functions that automatically morph their shapes over time and space in order to efficiently resolve all scales. Being mesh-free, the proposed method substantially reduces the computational cost as compared to existing adaptive methods. Furthermore, since the modes adapt themselves to the solution of the PDE, far fewer modes are needed to resolve all scales. This significantly reduces the memory requirements, thus outperforming the existing multiresolution methods.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.