  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.
