Title: Flexible spectral methods and high-level programming for PDEs
The large-scale numerical solution of PDEs is an essential part of scientific research. Decades of work have been put into developing fast numerical schemes for specific equations, but computational research in many fields is still largely software-limited. Here I will discuss how algorithmic flexibility and composability can enable new science, as illustrated by the Dedalus Project. Dedalus is an open-source Python framework that automates the solution of general PDEs using spectral methods. High-level abstractions allow users to symbolically specify equations, parallelize and scale their solvers to thousands of cores, and perform arbitrary analysis with the computed solutions. These features are enabling us to perform novel simulations of astrophysical and geophysical fluids with modern mathematical techniques. I will discuss applications using new bases for tensor-valued equations in spherical domains, immersed boundary methods for multiphase flows, and multi-domain simulations interfacing Dedalus with other PDE and integral equation solvers.
Keaton Burns is an applied math instructor at MIT working on scientific computing and fluid dynamics. He is developing the spectral PDE solver Dedalus and utilizing it to study diverse problems in astrophysical, geophysical, and biological fluids.