Cauchy-Lagrangian方法用于欧拉流的数值分析
The Cauchy-Lagrangian method for numerical analysis of Euler flow
AI总结：
本文提出一种新的半拉格朗日方法，用于求解理想不可压缩流体的欧拉方程，利用流体粒子轨迹的时间分析性，通过递推关系生成高阶时间泰勒系数，提升时间步长并提高计算效率与稳定性。
摘要：
本文介绍了一种新的半拉格朗日方法，用于数值求解理想不可压缩流体的欧拉方程，该方法利用流体粒子轨迹的时间分析性，并仅需初始数据有限的空间光滑性。通过从欧拉方程的Cauchy不变量公式推导出的递推关系，高效生成高阶时间泰勒系数。截断的高阶时间泰勒级数允许使用远超Courant-Friedrichs-Lewy限制的时间步长，而不会影响解的准确性。在二维欧拉方程上的测试表明，Cauchy-Lagrangian方法比欧拉ian Runge-Kutta方法更高效且更稳定，且比高阶欧拉ian时间泰勒算法更少受舍入误差快速增长的影响。我们还开发了适应Cauchy-Lagrangian方法的分析工具，如监测时间泰勒级数的收敛半径。某些其他流体方程也可以类似处理。
英文摘要：
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.
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