Windowed Fourier Methods for Overlapping Domain Approximations

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Partial differential equations on surfaces arise in many applications, including atmospheric research, evolution of coat patterns in animals, finger print formation, and deformation of elastic solids in fluids, to mention just a few. This research project aims to develop computationally efficient and accurate methods for numerical solution of such equations. The project is driven in particular by problems stemming from atmospheric models and will advance research on novel numerical schemes that are based on windowed Fourier approximations. Objectives include the design of algorithms for solving time-dependent differential equations on spherical geometries and other surfaces, the development of a rigorous mathematical analysis of convergence and stability for such algorithms, the implementation of high performance and scalable solvers, and to make efficient, accurate software available for use by the scientific community.Windowed Fourier methods are suitable for adaptive and parallel implementations and large-scale computations. The proposed schemes rely on domain decomposition, such as in the cubed sphere, with computations being carried out using fast Fourier transforms. Approximations are obtained on overlapping domains and a global solution is obtained by weighted average. Current high-order and spectral methods used in atmospheric research, such as spectral elements, are constrained by the well-known CFL condition; that is, node clustering in the space domain restricts the discretization in the time domain. Windowed Fourier methods use nearly uniform nodes on each subdomain, allowing for larger time steps when explicit schemes are used for time integration. For large-scale simulations, such as those required in climate prediction for example, each time step incurs significant communication cost at scale. By leveraging better node distributions, larger time steps and spectral accuracy, windowed Fourier schemes are expected to be more efficient than methods currently used in critical applications such as climate and geodynamics.