Solving Advection-Diffusion-Reaction Equations with Robin Boundary Conditions on Piecewise Smooth Interfaces

SSRN Electronic Journal · 2026-01-01

A review of level-set modeling in epitaxial growth and alloys solidification

Modelling and Simulation in Materials Science and Engineering · 2026-01-08

Abstract Level-set methods provide a powerful computational framework for simulating free boundary problems in materials science. This paper presents a unified perspective on their application to two distinct phenomena: multicomponent alloy solidification and epitaxial island growth. Although these problems differ significantly in physical mechanisms and characteristic length scales, they can both be effectively addressed within the level-set framework, highlighting the versatility of the method across diverse applications. We outline the mathematical formulations and highlight computational advances and common features across applications. This overview highlights the role of level-set methods as a foundational tool in predictive materials modeling.

Drag reduction in surfactant-contaminated superhydrophobic channels at high Péclet numbers

Journal of Fluid Mechanics · 2025-05-09 · 1 citations

Motivated by microfluidic applications, we investigate drag reduction in laminar pressure-driven flows in channels with streamwise-periodic superhydrophobic surfaces (SHSs) contaminated with soluble surfactant. We develop a model in the long-wave and weak-diffusion limit, where the streamwise SHS period is large compared with the channel height and the Péclet number is large. Using asymptotic and numerical techniques, we determine the influence of surfactant on drag reduction in terms of the relative strength of advection, diffusion, Marangoni effects and bulk–surface exchange. In scenarios with strong exchange, the drag reduction exhibits a complex dependence on the thickness of the bulk-concentration boundary layer and surfactant strength. Strong Marangoni effects immobilise the interface through a linear surfactant distribution, whereas weak Marangoni effects yield a quasi-stagnant cap. The quasi-stagnant cap has an intricate structure with an upstream slip region, followed by intermediate inner regions and a quasi-stagnant region that is mediated by weak bulk diffusion. The quasi-stagnant region differs from the immobile region of a classical stagnant cap, observed for instance in surfactant-laden air bubbles in water, by displaying weak slip. As exchange weakens, the bulk and interface decouple: the surfactant distribution is linear when the surfactant is strong, whilst it forms a classical stagnant cap when the surfactant is weak. The asymptotic solutions offer closed-form predictions of drag reduction across much of the parameter space, providing practical utility and enhancing understanding of surfactant dynamics in flows over SHSs.

Optimal control for stochastic neural oscillators

Biological Cybernetics · 2025-03-20 · 2 citations

This study develops an event-based, energy-efficient control strategy for desynchronizing coupled neuronal networks using optimal control theory. Inspired by phase resetting techniques in Parkinson's disease treatment, we incorporate stochasticity of the system's dynamics into deterministic models to address neural system intrinsic noise. We use an advanced computational solver for nonlinear stochastic partial differential equations to solve the stochastic Hamilton-Jacobi-Bellman equation via level set methods for a single neuron model; this allows us to find control inputs which drive the dynamics close to the system's phaseless set. When applied to coupled neuronal networks, these inputs achieve effective randomization of neuronal spike timing, leading to significant network desynchronization. Compared to its deterministic counterpart, our stochastic method can achieve considerable energy savings. The event-based control minimizes unnecessary charge transfer, potentially extending implanted stimulator battery life while maintaining robustness against variations in neuronal coupling strengths and network heterogeneities. These findings highlight the potential for developing energy-efficient neurostimulation techniques with implications for deep brain stimulation protocols. The presented computational framework could also be applied to other domains for which stochastic optimal control problems are prevalent.

Exogenous–Endogenous Surfactant Interaction Yields Heterogeneous Spreading in Complex Branching Networks

Physical Review Letters · 2025-01-23 · 3 citations

Experiments have shown that surfactant introduced to a liquid-filled maze can find the solution path. We reveal how the maze-solving dynamics arise from interactions between the added surfactant and endogenous surfactant present at the liquid surface. We simulate the dynamics using a nonlinear model solved with a discrete mimetic scheme on a graph. Endogenous surfactant transforms local spreading into a nonlocal problem with an omniscient view of the maze geometry, key to the maze-solving dynamics. Our results offer insight into surfactant-driven transport in complex networks such as lung airways.

CASL-HJX: A comprehensive guide to solving deterministic and stochastic hamilton-Jacobi equations

Computer Physics Communications · 2025-11-05

A Sharp Numerical Approach for Modeling Flows in Reactive Porous Media

SSRN Electronic Journal · 2025-01-01

CASL-HJX: A Comprehensive Guide to Solving Deterministic and Stochastic Hamilton-Jacobi Equations

ArXiv.org · 2025-05-12

CASL-HJX is a computational framework designed for solving deterministic and stochastic Hamilton-Jacobi equations in two spatial dimensions. It provides a flexible and efficient approach to modeling front propagation problems, optimal control problems, and stochastic Hamilton-Jacobi Bellman equations. The framework integrates numerical methods for hyperbolic PDEs with operator splitting techniques and implements implicit methods for second-order derivative terms, ensuring convergence to viscosity solutions while achieving global rather than local optimization. Built with a high-performance C++ core, CASL-HJX efficiently handles mixed-order derivative systems with time-varying dynamics, making it suitable for real-world applications across multiple domains. We demonstrate the solver's versatility through tutorial examples covering various PDEs and through applications in neuroscience, where it enables the design of energy-efficient controllers for regulating neural populations to mitigate pathological synchrony. While our examples focus on these applications, the mathematical foundation of the solver makes it applicable to problems in finance, engineering, and machine learning. The modular architecture allows researchers to define computational domains, configure problems, and execute simulations with high numerical accuracy. CASL-HJX bridges the gap between deterministic control methods and stochastic models, providing a robust tool for managing uncertainty in complex dynamical systems.

Drag reduction in surfactant-contaminated superhydrophobic channels at high Péclet numbers

arXiv (Cornell University) · 2024-06-21

Motivated by microfluidic applications, we investigate drag reduction in laminar pressure-driven flows in channels with streamwise-periodic superhydrophobic surfaces (SHSs) contaminated with soluble surfactant. We develop a model in the long-wave and weak-diffusion limit, where the streamwise SHS period is large compared to the channel height and the Péclet number is large. Using asymptotic and numerical techniques, we determine the influence of surfactant on drag reduction in terms of the relative strength of advection, diffusion, Marangoni effects and bulk-surface exchange. In scenarios with strong exchange, the drag reduction exhibits a complex dependence on the thickness of the bulk-concentration boundary layer and surfactant strength. Strong Marangoni effects immobilise the interface through a linear surfactant distribution, whereas weak Marangoni effects yield a quasi-stagnant cap. The quasi-stagnant cap has an intricate structure with an upstream slip region, followed by intermediate inner regions, and a quasi-stagnant region that is mediated by weak bulk diffusion. The quasi-stagnant region differs from the immobile region of a classical stagnant cap, observed for instance in surfactant-laden air bubbles in water, by displaying weak slip. As exchange weakens, the bulk and interface decouple: the surfactant distribution is linear when the surfactant is strong, whilst it forms a classical stagnant cap when the surfactant is weak. The asymptotic solutions offer closed-form predictions of drag reduction across much of the parameter space, providing practical utility and enhancing understanding of surfactant dynamics in flows over SHSs.

Preface

Communications on Applied Mathematics and Computation · 2024-04-10