Numerical methods for flow maps and potential applications to mixing and anomalous diffusion

In the work of V. Arnold (1966), it was shown that the incompressible Euler equations can be viewed as the L₂-geodesic equations on the Lie-group of volume preserving diffeomorphisms. This idea has since been used to study various fluid flows through their flow map. In this talk, I will present some recent works in computational fluid dynamics using the diffeomorphism group structure of the Euler equations. I will then propose some potential applications of these diffeomorphism-based numerical methods to the study of mixing enhancement and anomalous dissipation.

In the first part of the talk, I will present our work on the "characteristic mapping method", a numerical scheme for the Lie-advection equation based on computing the inverse Lagrangian flow map. This method has been applied to the linear advection and Euler equations in 2 and 3D. We achieve fully non-diffusive transport of advected quantities, where the numerical solutions are not truncated in frequency space.

I will present in the second part of the talk potential new numerical tools that could assist in the study of mixing enhancement and anomalous diffusion. In particular, using the advection flow map, the advection-diffusion equation can be pulled-back to Lagrangian frame to an anisotropic diffusion equation whose anisotropy depends on the pullback metric.