In this paper, we focus on numerical solutions for random genetic drift problem, which is governed by a degenerated convection-dominated parabolic equation. Due to the fixation phenomenon of genes, Dirac delta singularities will develop at boundary points as time evolves. Based on an energetic variational approach (EnVarA), a balance between the maximal dissipation principle (MDP) and least action principle (LAP), we obtain the trajectory equation. In turn, a numerical scheme is proposed using a convex splitting technique, with the unique solvability (on a convex set) and the energy decay property (in time) justified at a theoretical level. Numerical examples are presented for cases of pure drift and drift with semi-selection. The remarkable advantage of this method is its ability to catch the Dirac delta singularity close to machine precision over any equidistant grid.
Keywords: Random genetic drift, wright-fisher model, energetic variational approach, convex splitting scheme, Dirac delta singularity, fixation phenomenon
Duan, Chenghua 1 ; Liu, Chun 1 ; Wang, Cheng 1 ; Yue, Xingye 1
@article{M2AN_2019__53_2_615_0,
author = {Duan, Chenghua and Liu, Chun and Wang, Cheng and Yue, Xingye},
title = {Numerical complete solution for random genetic drift by energetic variational approach},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {615--634},
year = {2019},
publisher = {EDP Sciences},
volume = {53},
number = {2},
doi = {10.1051/m2an/2018058},
zbl = {1418.65150},
mrnumber = {3945576},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2018058/}
}
TY - JOUR AU - Duan, Chenghua AU - Liu, Chun AU - Wang, Cheng AU - Yue, Xingye TI - Numerical complete solution for random genetic drift by energetic variational approach JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2019 SP - 615 EP - 634 VL - 53 IS - 2 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2018058/ DO - 10.1051/m2an/2018058 LA - en ID - M2AN_2019__53_2_615_0 ER -
%0 Journal Article %A Duan, Chenghua %A Liu, Chun %A Wang, Cheng %A Yue, Xingye %T Numerical complete solution for random genetic drift by energetic variational approach %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2019 %P 615-634 %V 53 %N 2 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2018058/ %R 10.1051/m2an/2018058 %G en %F M2AN_2019__53_2_615_0
Duan, Chenghua; Liu, Chun; Wang, Cheng; Yue, Xingye. Numerical complete solution for random genetic drift by energetic variational approach. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 2, pp. 615-634. doi: 10.1051/m2an/2018058
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