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.

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.

DOI : 10.1051/m2an/2018058
Classification : 35K65, 92D10, 76M28, 76M30
Mots-clés : 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

1 
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     title = {Numerical complete solution for random genetic drift by energetic variational approach},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {615--634},
     publisher = {EDP-Sciences},
     volume = {53},
     number = {2},
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     url = {http://www.numdam.org/articles/10.1051/m2an/2018058/}
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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. http://www.numdam.org/articles/10.1051/m2an/2018058/

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