Applied Math Colloquium: Fruzsina Agocs
Friday, February 9, 2024 · 12 - 1 PM
Title: Chirps and waves: adaptive high-order methods for oscillatory ODEs and PDEs
Speaker: Fruzsina Agocs (Flatiron Institute, New York)
Abstract: Oscillatory problems have long posed a challenge to numerical computation, whether they take the form of ordinary or partial differential equations (ODEs or PDEs), yet they are ubiquitous in applications ranging from engineering to astrophysics. Oscillations force algorithms to use more discretization nodes (thus more computational effort) as the frequency grows. In PDEs, there are additional difficulties associated with waves interacting with the geometry, e.g. corners, periodic boundaries, cavities. In this talk, I will present two classes of fast, high-order accurate numerical methods to solve oscillatory problems. For ODEs, I introduce algorithms that exploit asymptotic expansions when the solution oscillates, but behave as ``standard'' solvers otherwise, thus achieving O(1) (frequency-independent) runtime. I will show how they eliminate computational bottlenecks in early-universe astrophysics. For PDEs, I will focus on two dimensional acoustic and electromagnetic scattering from a nonperiodic source by a periodic boundary. I show how these may be solved using boundary integral equation (BIE) methods, and by integrating over a family of quasiperiodic solutions. I will use this to discuss how dispersive trapped waves explain wave-guiding phenomena and an acoustic effect at Mayan pyramids.
