### 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.

Original language | English (US) |
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Pages (from-to) | 797-821 |

Number of pages | 25 |

Journal | BIT Numerical Mathematics |

Volume | 59 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Software
- Computer Networks and Communications
- Computational Mathematics
- Applied Mathematics

### Cite this

*BIT Numerical Mathematics*,

*59*(3), 797-821. https://doi.org/10.1007/s10543-019-00749-4

}

*BIT Numerical Mathematics*, vol. 59, no. 3, pp. 797-821. https://doi.org/10.1007/s10543-019-00749-4

**Behavior of different numerical schemes for random genetic drift.** / Xu, Shixin; Chen, Minxin; Liu, Chun; Zhang, Ran; Yue, Xingye.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Behavior of different numerical schemes for random genetic drift

AU - Xu, Shixin

AU - Chen, Minxin

AU - Liu, Chun

AU - Zhang, Ran

AU - Yue, Xingye

PY - 2019/9/1

Y1 - 2019/9/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85064353561&partnerID=8YFLogxK

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U2 - 10.1007/s10543-019-00749-4

DO - 10.1007/s10543-019-00749-4

M3 - Article

AN - SCOPUS:85064353561

VL - 59

SP - 797

EP - 821

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

SN - 0006-3835

IS - 3

ER -