Preserving Multiple Conserved Quantities of Stochastic Differential Equations via Projection Technique

Xuliang Li, Zhenyu Wang () and Xiaohua Ding
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Xuliang Li: School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China
Zhenyu Wang: Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China
Xiaohua Ding: Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China

Mathematics, 2025, vol. 13, issue 22, 1-20

Abstract: Stochastic differential equations (SDEs) with multiple conserved quantities are ubiquitous in scientific fields, modeling systems from molecular dynamics to celestial mechanics. While geometric numerical integrators that preserve single invariants are well-established, constructing efficient and high-order numerical schemes for SDEs with multiple conserved quantities remains a challenge. Existing approaches often suffer from high computational costs or lack desirable numerical properties like symmetry. This paper introduces two novel classes of projection-based numerical methods tailored for SDEs with multiple conserved quantities. The first method projects the increments of an underlying numerical scheme onto a discrete tangent space, ensuring all invariants are preserved by construction. The second method leverages a local coordinates approach, transforming the SDE onto the manifold defined by the invariants, solving it numerically, and then projecting back, guaranteeing the solution evolves on the correct manifold. We prove that both methods inherit the mean-square convergence order of their underlying schemes. Furthermore, we propose a simplified strategy that reduces computational expense by redefining the multiple invariants into a single one, offering a practical trade-off between exact preservation and efficiency. Numerical experiments confirm the theoretical findings and demonstrate the superior efficiency and structure-preserving capabilities of our methods.

Keywords: stochastic differential equations; multiple conserved quantities; projection; discrete gradient; local coordinates (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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