Numerical complete solution for random genetic drift by energetic variational approach
Numerical complete solution for random genetic drift by energetic variational approach
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DOI：
10.1051/m2an/2018058
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发表时间：
2018-03
期刊：
影响因子：
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通讯作者：
C. Duan;Chun Liu;Cheng Wang;Xingye Yue
中科院分区：
文献类型：
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作者：
C. Duan;Chun Liu;Cheng Wang;Xingye Yue
本文研究了一类退化对流主导抛物型方程的随机遗传漂变问题的数值解。由于基因的固定现象，随着时间的推移，狄拉克δ奇点会在边界点处出现。基于能量变分方法（EnVarA），平衡了最大耗散原理（MDP）和最小作用原理（LAP），得到了弹道方程。然后，利用凸分裂技术提出了一种数值格式，并在理论上证明了该格式的唯一可解性（在凸集上）和能量衰减性（在时间上）。给出了纯漂移和半选择漂移的数值算例。该方法的显著优点是能够在任何等距网格上捕捉接近机器精度的狄拉克三角洲奇点。
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.