Volume 53, Number 2, March-April 2019
|Page(s)||615 - 634|
|Published online||01 May 2019|
Numerical complete solution for random genetic drift by energetic variational approach
School of Mathematical Sciences, Soochow University, 215006 Suzhou, PR China
2 Department of Applied Mathematics, Illinois Institute of Technology, 60616 Chicago, IL, USA
3 Department of Mathematics, University of Massachusetts, Dartmouth, 02747-2300 North Dartmouth, `, USA
* Corresponding author: firstname.lastname@example.org
Accepted: 25 September 2018
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.
Mathematics Subject Classification: 35K65 / 92D10 / 76M28 / 76M30
Key words: Random genetic drift / wright-fisher model / energetic variational approach / convex splitting scheme / Dirac delta singularity / fixation phenomenon
© EDP Sciences, SMAI 2019
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