Stochastic Trajectories and Spectral Boundary Conditions for Enhanced Diffusion in Immersed Boundary Problems

Abstract: This work presents a comprehensive framework for enhanced diffusion modeling in fluid-structure interactions by combining the Immersed Boundary Method (IBM) with stochastic trajectories and high-order spectral boundary conditions. Using semi-Lagrangian schemes, this approach captures complex diffusion dynamics at moving interfaces, integrating probabilistic methods that reflect multi-scale fluctuations. In addition to a rigorous mathematical foundation that includes stability proofs, this model exhibits reduced numerical diffusion errors and improved stability in long-term simulations. Comparative studies highlight its effectiveness in multi-scale scenarios that require precision in interface dynamics. Focusing on various shear and circular flows, including those with H\"older and Lipschitz regularities and critical points, we establish sharp bounds on effective diffusion rates using specific initial data examples. This dual exploration in enhanced diffusion highlights how flow regularity and critical points influence dissipation. These findings advance both the theoretical understanding and practical applications of enhanced diffusion in fluid dynamics, offering new insights into diffusion rate optimization through interface dynamics and flow structure regularities. Future research can further refine the IBM framework by exploring alternative probabilistic methods to improve interface accuracy, opening up the potential for improved modeling in applications that require precise control over mixing rates and dissipation processes.