Strong convergence of a half-explicit Euler scheme for constrained stochastic mechanical systems

TL;DR

This paper proves strong convergence of a half-explicit Euler scheme that exactly preserves constraints in stochastic mechanical systems with nonlinear holonomic constraints, addressing a key challenge in numerical simulation.

Contribution

It introduces the first strong convergence analysis for a constraint-preserving scheme applied to nonlinear constrained stochastic mechanical systems.

Findings

Establishes pathwise uniform Lp-convergence of the scheme.

Provides a decomposition of discrete Lagrange multipliers for analysis.

Demonstrates the scheme's consistency, semi-stability, and moment growth properties.

Abstract

This paper is concerned with the numerical approximation of stochastic mechanical systems with nonlinear holonomic constraints. Such systems are described by second order stochastic differential-algebraic equations involving an implicitly given Lagrange multiplier process. The explicit representation of the Lagrange multiplier leads to an underlying stochastic ordinary differential equation, the drift coefficient of which is typically not globally one-sided Lipschitz continuous. We investigate a half-explicit drift-truncated Euler scheme which fulfills the constraint exactly. Pathwise uniform $L_p$-convergence is established. The proof is based on a suitable decomposition of the discrete Lagrange multipliers and on norm estimates for the single components, enabling the verification of consistency, semi-stability and moment growth properties of the scheme. To the best of our knowledge, the presented result is the first strong convergence result for a constraint-preserving scheme in the considered setting.