Parareal algorithms applied to stochastic differential equations with conserved quantities

TL;DR

This paper integrates projection methods into the parareal algorithm to efficiently solve stochastic differential equations while preserving conserved quantities, demonstrated through numerical experiments.

Contribution

It introduces a novel coupling of parareal algorithms with projection methods for conserved quantities in stochastic differential equations.

Findings

Convergence of the modified parareal algorithm is demonstrated.

Conservation of quantities is maintained in numerical simulations.

The approach improves solution accuracy for systems with invariants.

Abstract

In this papers, we couple the parareal algorithm with projection methods of the trajectory on a specific manifold, defined by the preservation of some conserved quantities of the differential equations. First, projection methods are introduced as the coarse and fine propagators. Second, we also apply the projection methods for systems with conserved quantities in the correction step of original parareal algorithm. Finally, three numerical experiments are performed by different kinds of algorithms to show the property of convergence in iteration, and preservation in conserved quantities of model systems.