Automating Differential Geometry For Fluid Mechanics

Abstract: Physical laws are coordinate-invariant, but practical computations are not. Good coordinates and frames can vastly simplify calculations, speeding up numerical simulations as well as facilitating rigorous proofs. But this doesn't make them easy to use. Though the chain rule and vector calculus are sufficient in principle, problems involving nonstandard geometries, curvilinear coordinates, or moving interfaces can rapidly spiral in complexity. Coordinate expansions become a major source of errors and lost time. Computer algebra systems help, but functionality for general geometries isn't "out-of-the-box". Other higher-level packages like xAct are powerful, but are often designed for index calculations in general relativity rather than typical continuum-mechanics PDEs with boundaries and constraints. There is a missing middle of tools to automatically convert systems of PDEs on nonstandard geometries to their concrete component forms.

I will introduce a small Mathematica package, Tensors, that aims to close this gap. Given coordinate mappings, tensor fields, frames, and metrics on manifolds and their boundaries, the package automatically translates expressions composed of standard differential operators and geometric data (e.g. gradients, integrals, curvatures, normals etc.) into component form. Automating this conversion makes complex geometric calculations faster, more reproducible, and easier to generalise. I will illustrate with an application from multiphase fluid dynamics, showing how the package streamlines geometric constructions needed for high-order asymptotics, leading to better-conditioned, complexity-optimal solvers for free-boundary problems.

Mathematicsfluid dynamics

Audience: researchers in the topic

Fluids and Structures Seminar @ UEA