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Numerical Solving of Smoluchowski Diffusion Equation in Two Dimensions
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This study developed and tested computer methods for solving a mathematical equation that describes how substances spread and move in two dimensions. The methods were found to be stable, accurate, and faster when run on specialized computer hardware, with potential future use in modeling how toxins attach to microplastic particles.
In this work we investigated the suitability of splitting methods using the Crank-Nicolson integrator for numerically solving the Smoluchowski diffusion equation in two dimensions. Smoluchowski diffusion equation is extension of diffusion equation, where chemical potential depends on the coordinates. Three numerical schemes were compared: Peaceman-Rachford Alternating Direction Implicit (ADI), Lie splitting with Crank-Nicolson, and Strang splitting with Crank-Nicolson. The splitting methods demonstrated superior stability and convergence properties compared to ADI, achieving absolute stability without the conditional stability limitations of explicit methods. Convergence analysis using the method of manufactured solutions confirmed second-order accuracy for the splitting schemes. Additionally computational parallelization was explored through GPU implementation, achieving speedups of up to twenty times for large grid sizes. The results establish splitting methods as robust and efficient approaches for solving two-dimensional Smoluchowski equations for practical systems. Applications of the developed methodology include predictions of local anesthetics pharmacokinetics, dynamics of extraction processes and absorption of lipophilic toxins to microplastics particles.
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This systematic review summarizes how mathematical models and computer simulations are being used to study microplastic pollution in the environment. These modeling tools help scientists predict where microplastics travel, how they accumulate, and where human exposure is most likely, which is crucial for developing effective strategies to protect public health.
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When a large batch of papers lands in the Atlas, we read through it and send a short write-up of what stood out.