An Eulerian variance-based finite-time Lyapunov exponent (vFTLE) approach for flows with uncertainties

We propose a novel partial differential equation (PDE) approach for computing the variance-based finite-time Lyapunov exponent (vFTLE) in stochastic vector fields. Our method modifies and extends finite-time variance analysis (FTVA) by incorporating the covariance matrix of the probability density function (PDF) associated with each initial takeoff location. This approach allows us to utilize the maximum eigenvalue of the covariance matrix to approximate the maximal stretching rate in uncertain flows. Additionally, we enhance computational efficiency by integrating stochastic sensitivity into an Eulerian framework, enabling the identification of regions with significant vFTLE values. This combination improves both the accuracy and efficiency of analyzing complex flow dynamics in stochastic environments.