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1. Passive concentration dynamics incorporated into the library IB2d, a two-dimensional implementation of the immersed boundary method

   https://doi.org/10.1088/1748-3190/ac4afa  

   Santiago, Matea; Battista, Nicholas A; Miller, Laura A; Khatri, Shilpa (March 2022, Bioinspiration & Biomimetics)

   Abstract In this paper, we present an open-source software library that can be used to numerically simulate the advection and diffusion of a chemical concentration or heat density in a viscous fluid where a moving, elastic boundary drives the fluid and acts as a source or sink. The fully-coupled fluid-structure interaction problem of an elastic boundary in a viscous fluid is solved using Peskin’s immersed boundary method. The addition or removal of the concentration or heat density from the boundary is solved using an immersed boundary-like approach in which the concentration is spread from the immersed boundary to the fluid using a regularized delta function. The concentration or density over time is then described by the advection-diffusion equation and numerically solved. This functionality has been added to our software library, IB2d , which provides an easy-to-use immersed boundary method in two dimensions with full implementations in MATLAB and Python. We provide four examples that illustrate the usefulness of the method. A simple rubber band that resists stretching and absorbs and releases a chemical concentration is simulated as a first example. Complete convergence results are presented for this benchmark case. Three more biological examples are presented: (1) an oscillating row of cylinders, representative of an idealized appendage used for filter-feeding or sniffing, (2) an oscillating plate in a background flow is considered to study the case of heat dissipation in a vibrating leaf, and (3) a simplified model of a pulsing soft coral where carbon dioxide is taken up and oxygen is released as a byproduct from the moving tentacles. This method is applicable to a broad range of problems in the life sciences, including chemical sensing by antennae, heat dissipation in plants and other structures, the advection-diffusion of morphogens during development, filter-feeding by marine organisms, and the release of waste products from organisms in flows. 

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2. On the Norm Equivalence of Lyapunov Exponents for Regularizing Linear Evolution Equations

   https://doi.org/10.1007/s00205-023-01928-y  

   Blumenthal, Alex; Punshon-Smith, Sam (October 2023, Archive for Rational Mechanics and Analysis)

   We consider the top Lyapunov exponent associated to a dissipative linear evolution equation posed on a separable Hilbert or Banach space. In many applications in partial differential equations, such equations are often posed on a scale of nonequivalent spaces mitigating, e.g., integrability (L^p) or differentiability (W^{s,p}). In contrast to finite dimensions, the Lyapunov exponent could a priori depend on the choice of norm used. In this paper we show that under quite general conditions, the Lyapunov exponent of a cocycle of compact linear operators is independent of the norm used. We apply this result to two important problems from fluid mechanics: the enhanced dissipation rate for the advection diffusion equation with ergodic velocity field; and the Lyapunov exponent for the 2d Navier–Stokes equations with stochastic or periodic forcing. 

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3. A Harris theorem for enhanced dissipation, and an example of Pierrehumbert ^*

   https://doi.org/10.1088/1361-6544/adc652  

   Cooperman, William; Iyer, Gautam; Son, Seungjae (April 2025, Nonlinearity)

   Abstract In many situations, the combined effect of advection and diffusion greatly increases the rate of convergence to equilibrium—a phenomenon known asenhanced dissipation. Here we study the situation where the advecting velocity field generates a random dynamical system satisfying certainHarris conditions. Ifκdenotes the strength of the diffusion, then we show that with probability at least $1-o\left({\kappa }^N\right)$enhanced dissipation occurs on time scales of order $|\ln \kappa |$, a bound which is known to be optimal. Moreover, on long time scales, we show that the rate of convergence to equilibrium is almost surelyindependentof diffusivity. As a consequence we obtain enhanced dissipation for the randomly shifted alternating shears introduced by Pierrehumbert’94. 

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4. Reaction diffusion system prediction based on convolutional neural network

   https://doi.org/10.1038/s41598-020-60853-2  

   Li, Angran; Chen, Ruijia; Farimani, Amir Barati; Zhang, Yongjie Jessica (March 2020, Scientific Reports)

   Abstract The reaction-diffusion system is naturally used in chemistry to represent substances reacting and diffusing over the spatial domain. Its solution illustrates the underlying process of a chemical reaction and displays diverse spatial patterns of the substances. Numerical methods like finite element method (FEM) are widely used to derive the approximate solution for the reaction-diffusion system. However, these methods require long computation time and huge computation resources when the system becomes complex. In this paper, we study the physics of a two-dimensional one-component reaction-diffusion system by using machine learning. An encoder-decoder based convolutional neural network (CNN) is designed and trained to directly predict the concentration distribution, bypassing the expensive FEM calculation process. Different simulation parameters, boundary conditions, geometry configurations and time are considered as the input features of the proposed learning model. In particular, the trained CNN model manages to learn the time-dependent behaviour of the reaction-diffusion system through the input time feature. Thus, the model is capable of providing concentration prediction at certain time directly with high test accuracy (mean relative error <3.04%) and 300 times faster than the traditional FEM. Our CNN-based learning model provides a rapid and accurate tool for predicting the concentration distribution of the reaction-diffusion system. 

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5. Error bounds for one-dimensional constrained Langevin approximations for nearly density-dependent Markov chains

   https://doi.org/10.1017/apr.2024.25  

   Campos, Felipe A; Williams, Ruth J (September 2024, Advances in Applied Probability)

   Abstract Continuous-time Markov chains are frequently used to model the stochastic dynamics of (bio)chemical reaction networks. However, except in very special cases, they cannot be analyzed exactly. Additionally, simulation can be computationally intensive. An approach to address these challenges is to consider a more tractable diffusion approximation. Leite and Williams (Ann. Appl. Prob.29, 2019) proposed a reflected diffusion as an approximation for (bio)chemical reaction networks, which they called the constrained Langevin approximation (CLA) as it extends the usual Langevin approximation beyond the first time some chemical species becomes zero in number. Further explanation and examples of the CLA can be found in Anderson et al.( SIAM Multiscale Modeling Simul.17, 2019). In this paper, we extend the approximation of Leite and Williams to (nearly) density-dependent Markov chains, as a first step to obtaining error estimates for the CLA when the diffusion state space is one-dimensional, and we provide a bound for the error in a strong approximation. We discuss some applications for chemical reaction networks and epidemic models, and illustrate these with examples. Our method of proof is designed to generalize to higher dimensions, provided there is a Lipschitz Skorokhod map defining the reflected diffusion process. The existence of such a Lipschitz map is an open problem in dimensions more than one. 

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