Behavior of different numerical schemes for population genetic drift problems

In this paper, we focus on numerical methods for the genetic drift problems, which 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. In order to find a \emph{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.Comment: 17 pages, 8 figure

A Comparative Study on Spin-Orbit Torque Efficiencies from W/ferromagnetic and W/ferrimagnetic Heterostructures

It has been shown that W in its resistive form possesses the largest spin-Hall ratio among all heavy transition metals, which makes it a good candidate for generating efficient dampinglike spin-orbit torque (DL-SOT) acting upon adjacent ferromagnetic or ferrimagnetic (FM) layer. Here we provide a systematic study on the spin transport properties of W/FM magnetic heterostructures with the FM layer being ferromagnetic Co$_{20}$Fe$_{60}$B$_{20}$ or ferrimagnetic Co$_{63}$Tb$_{37}$ with perpendicular magnetic anisotropy. The DL-SOT efficiency $|\xi_{DL}|$, which is characterized by a current-induced hysteresis loop shift method, is found to be correlated to the microstructure of W buffer layer in both W/Co$_{20}$Fe$_{60}$B$_{20}$ and W/Co$_{63}$Tb$_{37}$ systems. Maximum values of $|\xi_{DL}|\approx 0.144$ and $|\xi_{DL}|\approx 0.116$ are achieved when the W layer is partially amorphous in the W/Co$_{20}$Fe$_{60}$B$_{20}$ and W/Co$_{63}$Tb$_{37}$ heterostructures, respectively. Our results suggest that the spin Hall effect from resistive phase of W can be utilized to effectively control both ferromagnetic and ferrimagnetic layers through a DL-SOT mechanism