A novel semi-Lagrangian method is introduced to solve numerically the Euler equation for ideal incompressible flow in arbitrary space dimension. It exploits the time-analyticity of fluid particle trajectories and requires, in principle, only limited spatial smoothness of the initial data. Efficient generation of high-order time-Taylor coefficients is made possible by simple recurrence relations that follow from the Cauchy invariants formulation of the Euler equations (Zheligovsky & Frisch, J. Fluid Mech. 2014, vol. 749,404-430). Truncated time-Taylor series of very high order allow the use of time steps vastly exceeding the Courant-Friedrichs-Lewy limit, without compromising the accuracy of the solution.
Tests performed on the two-dimensional Euler equation indicate that the Cauchy--Lagrangian method is more - and occasionally much more - efficient and less prone to instability than Eulerian Runge-Kutta methods and less prone to rapid growth of rounding errors than the high-order Eulerian time-Taylor algorithm.
At the root of such results lies the geometrical content of the Euler equation as a geodesic flow on the (infinite-dimensional) manifold SDiff of measure-preserving diffeomorphisms. Indeed, the Cauchy invariants may be viewed as a consquence of Noether's theorem applied to the continuous relabelling symmetry. As to the stability of high-order time-Taylor expansions, it follows form Ebin and Marsden's (1970) observation that the geometric formulation of the Euler equation does not lose spatial derivatives.
Reference: O. Podvigina, V. Zheligovsky and U. Frisch. The Cauchy-Lagrangian method for numerical analysis of Euler flow.