Optimal Time Step Length for Lagrangian Interacting-Particle Simulations of Diffusive Mixing

Michael J. Schmidt; Nicholas B. Engdahl; David A. BensonDiogo Bolster

摘要

Abstract To date, Lagrangian mass-transfer-based interacting particle methods have been shown to be rigorous and capable of modeling a diverse range of sophisticated problems but have lacked formal criteria for choosing an optimal time step length (Δtdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$varDelta t$$end{document}). In Eulerian (grid-based) methods, a user can typically reduce Δtdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$varDelta t$$end{document} to an arbitrary level and expect to see corresponding gains in accuracy. The particle methods that we consider behave similarly, but only up to a point: for a fixed number of particles, Δtdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$varDelta t$$end{document} can become so small that the magnitude of diffusion restricts particles from communicating via mass-transfer, and at this point, solution accuracy begins to degrade. In this work, we formalize criteria for determining when this transition takes place, based on the properties of a particular system, and we use this criteria to choose the optimal Δtdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$varDelta t$$end{document}. We test these results with numerical experiments that demonstrate accurate prediction of the optimal Δtdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$varDelta t$$end{document} for a variety of conditions. Article Highlights We define the optimal time step length for mass-transfer particle tracking simulations.The guideline is shown to be effective for a variety of conditions.We provide guidance for determining the optimal time step for conditions different from those tested.

Optimal Time Step Length for Lagrangian Interacting-Particle Simulations of Diffusive Mixing

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