A Characteristic Mapping Method for the three-dimensional incompressible Euler equations

TL;DR

The paper introduces an efficient semi-Lagrangian characteristic mapping method for 3D incompressible Euler equations, accurately capturing vorticity and conserving energy over long times.

Contribution

It extends the 2D characteristic mapping method to 3D, utilizing a geometric approach based on differential forms and the Kelvin circulation theorem.

Findings

Method is globally third-order accurate.

Energy is conserved without artificial viscosity.

Small scales of the solution are preserved.

Abstract

We propose an efficient semi-Lagrangian Characteristic Mapping (CM) method for solving the three-dimensional (3D) incompressible Euler equations. This method evolves advected quantities by discretizing the flow map associated with the velocity field. Using the properties of the Lie group of volume preserving diffeomorphisms SDiff, long-time deformations are computed from a composition of short-time submaps which can be accurately evolved on coarse grids. This method is a fundamental extension to the CM method for two-dimensional incompressible Euler equations [51]. We take a geometric approach in the 3D case where the vorticity is not a scalar advected quantity, but can be computed as a differential 2-form through the pullback of the initial condition by the characteristic map. This formulation is based on the Kelvin circulation theorem and gives point-wise a Lagrangian description of the vorticity field. We demonstrate through numerical experiments the validity of the method and show that energy is not dissipated through artificial viscosity and small scales of the solution are preserved. We provide error estimates and numerical convergence tests showing that the method is globally third-order accurate.