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Applied Math Seminar
Moving Interfaces and Elliptic Systems
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Models of physical phenomena such as crystal growth or blood flow generally involve complex moving interfaces, with velocities determined by interfacial geometry and material physics. Numerical methods for such models tend to be customized. As a consequence, they must be redesigned whenever the model changes. We present a general computational algorithm for evolving complex interfaces which treats the velocity as a black box, thus avoiding model-dependent issues. The interface is implicitly updated via an explicit second-order semi-Lagrangian advection formula which converts moving interfaces to a contouring problem. Spatial and temporal resolutions are decoupled, permitting grid-free adaptive refinement of the interface geometry. Our modular implementation computes highly accurate solutions to geometric moving interface problems involving merging, anisotropy, faceting, curvature, dynamic topology and nonlocal interactions. For more general physical problems, the black-box interface velocity is often determined by a large elliptic system of PDEs. Fast new boundary integral techniques for general elliptic systems in complex domains are presented. These techniques, based on Fourier analysis and Ewald summation, yield a single model-independent implementation which solves moving interface problems as various as Stokes flow, Ostwald ripening and crystal growth. |