Near Preservation of Quadratic Invariants by Stochastic Runge-Kutta Methods

TL;DR

This paper analyzes conditions under which explicit stochastic Runge-Kutta methods can nearly preserve quadratic invariants, providing practical construction guidelines and error estimates, supported by numerical experiments.

Contribution

It introduces conditions for explicit SRK methods to preserve quadratic invariants and estimates errors from iterative implementations, advancing numerical methods for stochastic differential equations.

Findings

Explicit SRK methods can nearly preserve quadratic invariants.

Error estimates are provided for iterative implicit methods.

Numerical experiments confirm the theoretical preservation properties.

Abstract

Based on the combinatory theory of rooted colored trees, we investigate the conditions for the explicit stochastic Runge-Kutta (SRK) methods to preserve quadratic invariants (QI) up to certain orders of accuracy. These conditions can supply a practical approach of constructing explicit nearly conservative SRK methods. Meanwhile, we estimate errors in the preservation of QI resulting from iterative implementation of implicit conservative SRK methods with fixed-point and Newton's iterations. Finally, numerical experiments are performed to test the behavior of the methods in preserving QI.