Keaton Burns
Flatiron Institute

Title: Flexible algorithms for solving PDEs with spectral methods

Abstract: Global spectral methods are a powerful tool for solving PDEs in simple domains. They typically provide exponential convergence to smooth solutions as the discretization size is increased, and they can be easily applied to a wide range of equations. This talk will discuss generic algorithms for solving broad classes of PDEs using global spectral methods and their implementation in the open-source code Dedalus.

Dedalus is a Python-based PDE solver designed for maximum flexibility, incorporating features such as symbolic equation entry, custom domain construction, and automatic MPI parallelization. We will cover the key algorithmic features of the codebase, particularly how sparse Fourier and Chebyshev-tau methods can be used to discretize general equations in domains without coordinate singularities, and how new bases based on spin-weighted spherical harmonics and Jacobi polynomials can be used to sparsely discretize general tensorial equations in cylindrical and spherical domains.

Along the way, we will discuss examples of the code's capabilities with various applications to geophysical, astrophysical, and biological fluid dynamics, including efficient implementations of incompressible hydrodynamics without operator splitting in the context of turbulent glacial melting, implicitly timestepping sound waves in low Mach-number compressible flows, and simulating generalized Navier-Stokes models of active fluids.