From Kelvin and Helmholtz to Arnold, Khesin, and Moffatt, topology has drawn increased attention in fluid dynamics. Quantities such as helicity and enstrophy encode knotting, topological constraints, and fine structures such as turbulence energy cascades in both fluid and MHD systems. Several open scientific questions, such as corona heating, the generation of magnetic fields in astrophysical objects, and the Parker hypothesis, call for topology-preserving computation. 

 

In this talk, we investigate the role of topology (knots and cohomology) in computational fluid dynamics by two examples: relaxation and dynamo. We investigate the question of “why structure-preservation” in this context and discuss some recent results on topology-preserving numerical analysis and computation. Finite Element Exterior Calculus sheds light on tackling some long-standing challenges and establishing a computational approach for topological (magneto)hydrodynamics.