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Xingye Yue

2013-12-26
11:10:00 - 12:00:00

101 , Mathematics Research Center Building (ori. New Math. Bldg.)

We focus on numerical methods to solve the diffusion equation for the random genetic drift that occurs at a single unlinked locus with two alleles. It is a degenerated convection-dominated parabolic equation. Two finite volume methods, upwind (UFVM) and central (CFVM) schemes are used to solve the equation numerically. We observed that the long time behaviors of the numerical solutions of these methods are totally different. Based on the conservations of total probability and the mean gene frequency (expectation), the conclusion is drawn that the results of UFVM are not correct since it destroys the conservation of the mean gene frequency. However, in general, the upwind scheme is a better choice for the convection-dominated problem to achieve stability due to its intrinsic numerical viscosity. To see what’s wrong here, we appeal to the method of vanishing viscosity, i.e., a small viscosity term is first added in, then the limit behavior of the solution is investigated when the added viscosity tends to zero. We see that the limitation of the steady state solution is uniquely determined and has nothing to do with the initial conditions. This means that the long time behavior of the original problem will be changed by any added infinitesimal viscosity. That is the reason why the upwind scheme does not work for the genetic drift problem.
Furthermore, we have to answer another question, why central scheme works well for a convection-dominated problem? To this purpose, we find that convection should be classified into 2 types: one is related to the stability of the numerical scheme and the other has nothing to do with the stability of the numerical scheme, no matter whether it is dominated. This is totally a new observation for convection-diffusion community.
This is a joint work with Minxin Chen, Chun Liu, David Waxman and Shixin Xu.

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