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A novel semi-Lagrangian method is introduced to solve numerically the Eulerequation for ideal incompressible flow in arbitrary space dimension. Itexploits the time-analyticity of fluid particle trajectories and requires, inprinciple, only limited spatial smoothness of the initial data. Efficientgeneration of high-order time-Taylor coefficients is made possible by arecurrence relation that follows from the Cauchy invariants formulation of theEuler equation (Zheligovsky & Frisch, J. Fluid Mech. 2014, 749, 404-430).Truncated time-Taylor series of very high order allow the use of time stepsvastly exceeding the Courant-Friedrichs-Lewy limit, without compromising theaccuracy of the solution. Tests performed on the two-dimensional Euler equationindicate 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 Euleriantime-Taylor algorithm. We also develop tools of analysis adapted to theCauchy-Lagrangian method, such as the monitoring of the radius of convergenceof the time-Taylor series. Certain other fluid equations can be handledsimilarly.

The Cauchy-Lagrangian method for numerical analysis of Euler flow