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

TL;DR

This paper introduces a semi-Lagrangian method utilizing the gradient-augmented level set approach for high-precision, efficient simulation of 2D incompressible Euler equations on coarse grids, with controlled error and conservation properties.

Contribution

It presents a novel flow map evolution technique with error control via submap decomposition, achieving exponential resolution in linear time.

Findings

High accuracy demonstrated on vortex merger and flow simulations

Method outperforms traditional Cauchy-Lagrangian approaches

Error can be effectively controlled through remapping times

Abstract

We propose an efficient semi-Lagrangian method for solving the two-dimensional incompressible Euler equations with high precision on a coarse grid. The new approach evolves the flow map using the gradient-augmented level set method (GALSM). Since the flow map can be decomposed into submaps (each over a finite time interval), the error can be controlled by choosing the remapping times appropriately. This leads to a numerical scheme that has exponential resolution in linear time. Error estimates are provided and conservation properties are analyzed. The computational efficiency and the high precision of the method are illustrated for a vortex merger and a four mode and a random flow. Comparisons with a Cauchy-Lagrangian method are also presented.