TY - JOUR
T1 - Numerical complete solution for random genetic drift by energetic variational approach
AU - Duan, Chenghua
AU - Liu, Chun
AU - Wang, Cheng
AU - Yue, Xingye
N1 - Funding Information:
Acknowledgements. It is grateful to Prof. Xinfu Chen for helpful discussions. This work is supported in part by NSF of China under the grants 11271281. Chun Liu is supported by NSF-DMS 1759535 and NSF-DMS 1759536. And Cheng Wang is supported by NSF grants DMS-1418689.
Publisher Copyright:
© EDP Sciences, SMAI 2019
PY - 2019
Y1 - 2019
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85062151771&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85062151771&partnerID=8YFLogxK
U2 - 10.1051/m2an/2018058
DO - 10.1051/m2an/2018058
M3 - Article
AN - SCOPUS:85062151771
SN - 2822-7840
VL - 53
SP - 615
EP - 634
JO - ESAIM: Mathematical Modelling and Numerical Analysis
JF - ESAIM: Mathematical Modelling and Numerical Analysis
IS - 2
ER -