The Cauchy-Lagrangian method for numerical analysis of Euler flow

TL;DR

The paper introduces a high-order semi-Lagrangian numerical method for Euler flow that leverages time-analyticity of trajectories, enabling larger time steps and improved stability over traditional Eulerian methods.

Contribution

A new Cauchy-Lagrangian method utilizing high-order time-Taylor series and recurrence relations for efficient, stable simulation of ideal incompressible Euler flows in any dimension.

Findings

More efficient than Eulerian Runge-Kutta methods.

Less prone to rounding errors than high-order Eulerian time-Taylor algorithms.

Enables larger time steps without loss of accuracy.

Abstract

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 a recurrence relation that follows from the Cauchy invariants formulation of the Euler equation (Zheligovsky & Frisch, J. Fluid Mech. 2014, 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. We also develop tools of analysis adapted to the Cauchy-Lagrangian method, such as the monitoring of the radius of convergence of the time-Taylor series. Certain other fluid equations can be handled similarly.