2025/03/24 by Hudson, Thomas, Li, Xingjie Helen, Murphy, Sarah
#60H35 #65L20 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2503.19203
In this paper, we revisit the numerical stability of four well-established explicit stochastic integration schemes through a new generic benchmark stochastic differential equation (SDE) designed to assess asymptotic statistical accuracy and stability properties. This one-parameter benchmark equation is derived from a general one-dimensional first-order SDE using spatio-temporal nondimensionalization and is employed to evaluate the performance of (1) Euler-Maruyama (EM), (2) Milstein (Mil), (3) Stochastic Heun (SH), and (4) a three-stage Runge-Kutta scheme (RK3). Our findings reveal that lower-order schemes can outperform higher-order ones over a range of time step sizes, depending on the benchmark parameters and application context. The theoretical results are validated through a series of numerical experiments, and we discuss their implications for more general applications, including a nonlinear example of particle transport in porous media under various conditions. Our results suggest that the insights obtained from the linear benchmark problem provide reliable guidance for time-stepping strategies when simulating nonlinear SDEs.