Easy-to-Implement One-Step Schemes for Stochastic Integration

TL;DR

The paper develops an easily implementable framework to convert deterministic one-step integrators into stochastic schemes for Stratonovich SDEs, enabling SRK, SIFRK, and SETDRK methods with minimal modification to existing solvers. A fundamental convergence theorem links deterministic order $p$ to stochastic strong orders, capturing symmetric Stratonovich terms up to $p/2$ and improving order under commutativity conditions. Through extensive numerical experiments on multidimensional and commutative noise, as well as drift-commutative scenarios and a 2D Navier–Stokes SPDE, the authors show SETDRK and SIFRK often outperform SRK at large timesteps and in low-noise regimes, while maintaining deterministic accuracy when noise vanishes. The work demonstrates practical impact for ensemble forecasting and uncertainty quantification by enabling larger timesteps and improved efficiency with structured stochastic integrators.

Abstract

Convenient, easy to implement stochastic integration methods are developed on the basis of abstract one-step deterministic order $p$ integration techniques. The abstraction as an arbitrary one step map allows the inspection of easy to implement stochastic exponential time differencing Runge-Kutta (SETDRK), stochastic integrating factor Runge-Kutta (SIFRK) and stochastic RK (SRK) schemes. Such schemes require minimal modifications to existing deterministic schemes and converging to the Stratonovich SDE, whilst inheriting many of their desirable properties. These schemes capture all symmetric terms in the Stratonovich-Taylor expansion, are order $p$ in the limit of vanishing noise, can attain at least strong order $p/2$ or $p/2-1/2$ (parity dependent) for drift commutative noise, strong order $1$ for commutative noise, and strong order $1/2$ for multidimensional non-commutative noise. Numerical convergence is demonstrated using different bases of noise for 2nd, 3rd and 4th order SETDRK, SIFRK and SRK schemes for a stochastic KdV equation.

Figures (11)

• Figure 1.1: Waterfall Plots of deterministic and stochastic KdV and KS equations under nonlinear transport noise.
• Figure 4.1: Non-commutative basis $\lbrace\xi_{m}\rbrace_{m=1,2,3}$
• Figure 4.2: Temporal convergence for non-commutative noise. In all schemes we see that for non-commutative noise, we attain strong order $1/2$ in the limit of $\Delta t\rightarrow 0$ in \ref{['fig:all order non-commutative basis']}. SETDTK and SIFRK methods could be computed at timesteps larger than RK methods. SETDTK had errors slightly lower than SIFRK methods. In \ref{['fig:all order non-commutative basis']} when using larger timesteps there was significant practical merit in using higher order schemes.
• Figure 4.3: CPU-time vs Relative error
• Figure 4.4: Commutative basis $\lbrace\xi_{m}\rbrace_{m=1,2,3}$.

Theorems & Definitions (15)

• Definition 2.1: Mean-square convergence
• Theorem 2.1: FTMSC milstein2004stochastic
• Definition 2.2: $L^0$, $L^i$ operators
• Definition 2.3: Bracket
• Definition 2.4: Commutative
• Definition 2.5: Drift Commutative
• Theorem 2.2
• proof
• Remark
• Definition 2.6