Xingye Yue, School of Math Sciences, Soochow University
601, Pao Yue-Kong Library
We focus on numerical methods for the Wright-Fisher model on the random genetic drift that occurs at a single unlinked locus with two or more alleles. It is governed by a degenerated convection-dominated parabolic equation. Due to the degeneration and convection, Dirac singularities will always be developed at boundary points as time evolves, which is just the so-called fixation phenomenon. In order to find a complete solution which should keep the conservation of total probability and expectation, three different schemes based on finite volume methods are used to solve the equation numerically: one is a upwind scheme, the other two are different central schemes. We observed that all the methods are stable and can keep the total probability, but have totally different long-time behaviors concerning with the conservation of expectation. We prove that any extra infinitesimal diffusion leads to a same artificial steady state. So upwind scheme does not work due to its intrinsic numerical viscosity. We find one of the central schemes introduces a numerical viscosity term too, which is beyond the common understanding in the convection-diffusion community. Careful analysis is presented to prove that the other central scheme does work. Our study shows that the numerical methods should be carefully chosen and any method with intrinsic numerical viscosity must be avoided. Numerical methods for multi-dimensional Wright-Fisher model are also discussed.
This is a joint work with Xinfu Chen, Chun Liu, David Waxman and Shixin Xu.