On fast computation of finite-time coherent sets using radial basis functions

Area of Science:

• Fluid dynamics
• Dynamical systems theory
• Numerical analysis

Background:

• Finite-time coherent sets are crucial for understanding and predicting mixing phenomena in fluid systems.
• The transfer operator method is a powerful tool for detecting these sets, but its computational cost, particularly operator construction, can be prohibitive.
• Existing methods often require extensive computation of Lagrangian trajectories.

Purpose of the Study:

• To present a novel, computationally efficient numerical method for constructing transfer operators to detect finite-time coherent sets.
• To specifically address the computational bottleneck in transfer operator analysis for advective dynamics.
• To enable faster and more scalable analysis of Lagrangian coherent structures.

Main Methods:

• Radial basis function collocation for numerical approximation.
• Application to a dynamic isoperimetric transfer operator construction designed for advective flows.
• Minimization of coherent set boundary size relative to volume using a dynamic Laplace operator.

Main Results:

• A substantial reduction in the number of required Lagrangian trajectories.
• Significant speedups in transfer operator analysis compared to previous methods.
• Efficient detection of finite-time coherent sets in advective systems.

Conclusions:

• The proposed numerical method offers a computationally advantageous alternative for identifying finite-time coherent sets.
• This advancement facilitates more efficient analysis of Lagrangian coherent structures, particularly in computationally intensive scenarios.
• The method enhances the practical applicability of transfer operator techniques in fluid dynamics and related fields.