A characteristic mapping method for two-dimensional incompressible Euler flows

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. The computational efficiency and the high precision of the method are illustrated for a vortex merger and a four mode flow. Comparisons with a Cauchy-Lagrangian method are also presented.