This paper will investigate the dynamics of numerical methods for solving stochastic differential equations (SDEs) based on two different approaches, the stability of the numerical approximation and the stationary solution of the Fokker-Planck equation of the underlying SDEs respectively. The first approach will involve the use of fixed-point iteration and the distribution theory to determine how the parameters in the problem and the method(stepsize) can be chosen in order to get a stable solution with higher probability. Meanwhile, by using Fokker-Planck equation, the qualitative behavior of the stationary probability density function can be obtained. These two different approaches give similar interpretation on the dynamics of the solution. The problem that we will consider comes from population dynamics and is based on the Verhulst (logistic) equation. In particular, we will consider both the additive and multiplicative forms of the problem. We will consider the dynamics of The Euler-Maruyama method (strong order 0.5) and the so-called R2 method (strong order 1) as applied to this problem.

MINISIMPOSIUM SESSION: 7. Numerical methods for Stochastic equations