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Behavior of different numerical schemes for random genetic drift

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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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Acknowledgements

The authors benefitted a great deal from discussions with Prof. David Waxman, Prof. Xinfu Chen and Prof. Xiaobing Feng. The authors thank the anonymous referees for their most valuable comments which improve the paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Soochow University, Suzhou, 215006, China

    Shixin Xu, Minxin Chen & Xingye Yue

  2. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, 60616, USA

    Chun Liu

  3. School of Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Ran Zhang

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  1. Shixin Xu
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  2. Minxin Chen
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  3. Chun Liu
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  4. Ran Zhang
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  5. Xingye Yue
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Correspondence to Xingye Yue.

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Communicated by Elisabeth Larsson.

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Partially supported by NSF of China Grants 11271281, 11301368 and 91230106 and by NSF Grants DMS-1159937, DMS-1216938 and DMS-1109107.

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Xu, S., Chen, M., Liu, C. et al. Behavior of different numerical schemes for random genetic drift. Bit Numer Math 59, 797–821 (2019). https://doi.org/10.1007/s10543-019-00749-4

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