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B-series for SDEs with application to exponential integrators for non-autonomous semi-linear problems

Alemayehu Adugna Arara, Kristian Debrabant, Anne Kværnø

TL;DR

This work consolidates stochastic B-series theory for SDEs and extends it to non-autonomous and semi-linear problems, enabling systematic order analysis and the design of exponential integrators. By formulating exact and numerical solutions as B-series for partitioned, non-autonomous, and semi-linear SDEs, it provides composition and product rules that hold without requiring Itô/Stratonovich restrictions or linear commutativity. The authors develop a unified framework with Lawson-type transformations and extended tree sets to capture deterministic, stochastic, and time-dependent components, facilitating the analysis and construction of exponential Runge–Kutta methods. The approach yields practical order-condition tools and is demonstrated through concrete examples, including an exponential midpoint rule, illustrating applicability to non-autonomous, non-commuting linear parts and time-dependent diffusion.

Abstract

In this paper a set of previous general results for the development of B--series for a broad class of stochastic differential equations has been collected. The applicability of these results is demonstrated by the derivation of B--series for non-autonomous semi-linear SDEs and exponential Runge-Kutta methods applied to this class of SDEs, which is a significant generalization of existing theory on such methods.

B-series for SDEs with application to exponential integrators for non-autonomous semi-linear problems

TL;DR

This work consolidates stochastic B-series theory for SDEs and extends it to non-autonomous and semi-linear problems, enabling systematic order analysis and the design of exponential integrators. By formulating exact and numerical solutions as B-series for partitioned, non-autonomous, and semi-linear SDEs, it provides composition and product rules that hold without requiring Itô/Stratonovich restrictions or linear commutativity. The authors develop a unified framework with Lawson-type transformations and extended tree sets to capture deterministic, stochastic, and time-dependent components, facilitating the analysis and construction of exponential Runge–Kutta methods. The approach yields practical order-condition tools and is demonstrated through concrete examples, including an exponential midpoint rule, illustrating applicability to non-autonomous, non-commuting linear parts and time-dependent diffusion.

Abstract

In this paper a set of previous general results for the development of B--series for a broad class of stochastic differential equations has been collected. The applicability of these results is demonstrated by the derivation of B--series for non-autonomous semi-linear SDEs and exponential Runge-Kutta methods applied to this class of SDEs, which is a significant generalization of existing theory on such methods.
Paper Structure (5 sections, 7 theorems, 52 equations)

This paper contains 5 sections, 7 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Stochastic B--series: Definitions and main results
  3. Non-autonomous SDEs
  4. Semi-linear SDEs and exponential integrators
  5. Example 2 applied to the SDE in Example 1

Key Result

Lemma 1

If ${V}^{(q)}(h) = B^{(q)}(\phi,x_0;h)$, $q=1,\dots,Q$, are some B-series with $\phi(\emptyset_q)\equiv 1$ and $f \in C^{\infty}(\mathbb{R}^{d_1}\times \ldots \times \mathbb{R}^{d_Q}, \mathbb{R}^{d})$, then $f({V}^{(1)}(h),\dots,{V}^{(Q)}(h))$ can be written as a formal series of the form where

Theorems & Definitions (18)

  • Example 1
  • Definition 1: Trees and combinatorial coefficients
  • Definition 2: Elementary differentials
  • Lemma 1
  • Theorem 1
  • Definition 3: Tree order
  • Example 2
  • Theorem 2: Composition of B--series
  • Lemma 2
  • Example 3
  • ...and 8 more