## 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 |

## All Science Journal Classification (ASJC) codes

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