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

•The method achieves third-order global convergence as derived in section 3.3.1 and demonstrated numerically in section 3.3.2.•The method provides an accurate conservation of all vorticity moments (Corollary 1, section 3.3.1), and e.g. 4.1 and 4.2 (Tables 2 and 4).•The method has long-time stability. Section 4.1 and 4.2 carry out the same tests featured in [15]. Additionally, similar tests are run to longer times with no issues.•The Characteristic Mapping method provides arbitrary subgrid resolution of the solution. We provide a 8192X zoom into the solution in Figs. 7, 8, 9 and 10. We propose an efficient semi-Lagrangian method for solving the two-dimensional incompressible Euler equations with high precision on a coarse grid. This new approach evolves the flow map using a combination of the Characteristic Mapping (CM) method [1] the gradient-augmented level-set (GALS) method [2]. The flow map possesses a semigroup structure which allows for the decomposition of a long-time deformation into short-time submaps. This leads to a numerical scheme that achieves exponential resolution in linear time. Error estimates are provided and conservation properties are analysed. The computational efficiency and the high precision of the method are illustrated in the vortex merger, four-modes and random flow problems. Comparisons with the Cauchy-Lagrangian method [3] are also presented.