Behavior of different numerical schemes for random genetic drift

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2019-09-01

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Abstract

In the problem of random genetic drift, the probability density of one gene is governed by a degenerated convection-dominated diffusion equation. Dirac singularities will always be developed at boundary points as time evolves, which is known as the fixation phenomenon in genetic evolution. Three finite volume methods: FVM1-3, one central difference method: FDM1 and three finite element methods: FEM1-3 are considered. These methods lead to different equilibrium states after a long time. It is shown that only schemes FVM3 and FEM3, which are the same, preserve probability, expectation and positiveness and predict the correct probability of fixation. FVM1-2 wrongly predict the probability of fixation due to their intrinsic viscosity, even though they are unconditionally stable. Contrarily, FDM1 and FEM1-2 introduce different anti-diffusion terms, which make them unstable and fail to preserve positiveness.

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Science & Technology, Technology, Physical Sciences, Computer Science, Software Engineering, Mathematics, Applied, Computer Science, Mathematics, Random genetic drift, Degenerate equation, Conservations of probability and expectation, Finite volume method, Finite difference method, Finite element method, Numerical viscosity and numerical anti-diffusion, DIFFUSION EQUATION, FIXATION

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https://hdl.handle.net/10161/27448

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10.1007/s10543-019-00749-4

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Xu, S, M Chen, C Liu, R Zhang and X Yue (2019). Behavior of different numerical schemes for random genetic drift. BIT Numerical Mathematics, 59(3). pp. 797–821. 10.1007/s10543-019-00749-4 Retrieved from https://hdl.handle.net/10161/27448.

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Xu

Shixin Xu

Assistant Professor of Mathematics at Duke Kunshan University

Shixin Xu is an Assistant Professor of Mathematics whose research spans several dynamic and interconnected fields. His primary interests include machine learning and data-driven models for disease prediction, multiscale modeling of complex fluids, neurovascular coupling, homogenization theory, and numerical analysis. His current projects reflect a diverse and impactful portfolio:

  • Developing predictive models based on image data to identify hemorrhagic transformation in acute ischemic stroke.
  • Conducting electrodynamics modeling of saltatory conduction along myelinated axons to understand nerve impulse transmission.
  • Engaging in electrochemical modeling to explore the interactions between electric fields and chemical processes.
  • Investigating fluid-structure interactions with mass transport and reactions, crucial for understanding physiological and engineering systems.

These projects demonstrate his commitment to addressing complex problems through interdisciplinary approaches that bridge mathematics with biological and physical sciences.

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