Numerical Stability Revisited: A Family of Benchmark Problems for the Analysis of Explicit Stochastic Differential Equation integrators

TL;DR

The 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, and suggest that the insights obtained from the linear benchmark problem provide reliable guidance for time-stepping strategies when simulating nonlinear SDEs.

Abstract

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.

Figures (4)

• Figure 1: Comparing 1st Moment evolutions up to time $T=20$ among the analytic and numerical schemes. The first row depicts when $\eta=0.1$ (Eta) for stepsizes $h=0.001,0.01,0.1,$ and the second row depicts when $\eta=1.4$ with the same range of stepsizes. The moment calculations on the $y$-axis are plotted on a logarithmic scale. We see that the EM and Milstein schemes are asympototically first moment stable and accurate, whereas RK3 and SH are biased.
• Figure 2: Comparing 2nd Moment evolutions up to time $T=20$ between analytic and numerical schemes. The first row depicts when $\eta=0.1$ (Eta) for stepsizes $h=0.001,0.01,0.1,$ and the second row depicts when $\eta=1.4$ with the same range of stepsizes. These plots show that all schemes are asympotically stable in the second moment, with Milstein being the most biased.
• Figure 3: Subfigs 1--4: 2nd moment stability regions (pink filled) and 1st moment stability regions (lines below) for EM, Milstein, SH and RK3 schemes. Subfig 5: comparison of 2nd moment stability regions among all schemes. Regarding the 1st moment: EM and Milstein are more stable. Regarding the 2nd moment: RK3 is more stable when $\eta\le 0.5$ or $\eta >1.145$; and EM is more stable when $0.5<\eta<1.145$.
• Figure 4: Numerical comparison for stochastic simulation of 1D porous medium \ref{['Porous_SDE']}. The velocity and diffusion coefficients and their linear approximations of \ref{['porus1d_vel_diff']} are plotted in the first and second columns. Fine (red) and coarse (blue) mean path and their absolute discrepancies are plotted in the third and forth columns. Top row is for $\eta<0.5$ and the bottom row is for $\eta>0.5$.

Theorems & Definitions (3)

• Remark 3.1
• Remark 3.2
• Definition 4.1