Lawson schemes for highly oscillatory stochastic differential equations and conservation of invariants

TL;DR

This paper develops stochastic Lawson integrating-factor schemes for highly oscillatory Stratonovich SDEs with linear and nonlinear drift and diffusion. By transforming X(t) via an exponential of the linear part, it derives implicit midpoint and trapezoidal Lawson schemes and proves that quadratic invariants, $\,\mathcal{I}(X)=X^T D X$, are preserved under the midpoint variant under standard skew-symmetry and commutativity assumptions, with near-preservation for the trapezoidal variant when diffusion terms are fully included in the exponent. The authors establish strong and weak convergence properties of SL schemes and demonstrate, through numerical experiments on stochastic rigid body, non-linear Kubo oscillator, and stochastic FPUT problems, that Lawson schemes better resolve fast oscillations and can maintain invariants more faithfully than standard methods, with FSL schemes offering advantages in high-noise regimes. These results indicate that stochastic Lawson methods provide a robust and efficient framework for simulating highly oscillatory SDEs in applied contexts.

Abstract

In this paper, we consider a class of stochastic midpoint and trapezoidal Lawson schemes for the numerical discretization of highly oscillatory stochastic differential equations. These Lawson schemes incorporate both the linear drift and diffusion terms in the exponential operator. We prove that the midpoint Lawson schemes preserve quadratic invariants and discuss this property as well for the trapezoidal Lawson scheme. Numerical experiments demonstrate that the integration error for highly oscillatory problems is smaller than that of some standard methods.

Figures (15)

• Figure 1: Numerical trajectory of the non-linear Kubo oscillator. Blue: reference solution, blue circle: value at $t=1$. Red triangles: numerical approximation, red circle: value at $t=1$.
• Figure 2: The stochastic rigid body problem: Strong error vs. step size for different values of $\sigma$ and $\omega$.
• Figure 3: The stochastic rigid body problem: Wall-clock time per batch of 25 paths vs. accuracy for different values of $\sigma$ and $\omega$.
• Figure 4: The stochastic rigid body problem: Weak error of $I$ vs. step size for different values of $\sigma$ and $\omega$.
• Figure 5: The stochastic rigid body problem: The invariant $\mathcal{I}(Y_n)$.

Theorems & Definitions (12)

• Example 1: The non-linear Kubo oscillator
• Theorem 1: Convergence of stochastic Lawson methods debrabant21rkl
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 2
• Corollary 1
• proof : of \ref{['thm:NumQuadInv']}