Numerical complete solution for random genetic drift by energetic variational approach

¹ 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: xyyue@suda.edu.cn

Received: 20 August 2017
Accepted: 25 September 2018

Abstract

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.