LTB: Accurate Computational Methods for Very Large Dynamical Systems

动力系统理论的基本原理，如闭合引理和庞加莱递归定理，表明动力学可以用周期运动来理解。该项目将使用这些原理作为灯塔，并开发数值方法，通过精确计算稳定解、行波、周期运动和相对周期运动来阐明具有超过一百万自由度的系统的动力学。这些方法将应用于低和中等雷诺数下的流体湍流，以及技术上重要的向湍流的过渡问题。本项目将通过训练学生熟练使用和创建科学软件来推进计算科学的教学。

Foundational principles of the theory of dynamical systems, such as the closing lemma and the Poincare recurrence theorem, suggest that dynamics can be understood quite generally in terms of periodic motions. This project will use those principles as a beacon and develop numerical methods that elucidate the dynamics of systems with more than a million degrees of freedom using accurate computations of steady solutions, traveling waves, periodic motions, and relative periodic motions. These methods will be applied to fluid turbulence at low and moderate Reynolds numbers and to the technologically important transition-to-turbulence problem. This project will advance the instruction of computational science by training students to be skillful at using and creating scientific software.