Novel semi-explicit symplectic schemes for nonseparable stochastic Hamiltonian systems

TL;DR

This paper introduces new semi-explicit symplectic schemes for nonseparable stochastic Hamiltonian systems that improve efficiency and accuracy, especially for stochastic Schrödinger equations, by preserving invariants and using symmetric projection.

Contribution

The paper develops novel stochastic semi-explicit symplectic schemes with augmented Hamiltonians and symmetric projection, enhancing performance for nonseparable systems.

Findings

Outperforms traditional stochastic midpoint scheme in effectiveness.

Preserves quadratic invariants in certain cases.

Efficient Newton iteration solver simplifies implementation.

Abstract

In this manuscript, we propose efficient stochastic semi-explicit symplectic schemes tailored for nonseparable stochastic Hamiltonian systems (SHSs). These semi-explicit symplectic schemes are constructed by introducing augmented Hamiltonians and using symmetric projection. In the case of the artificial restraint in augmented Hamiltonians being zero, the proposed schemes also preserve quadratic invariants, making them suitable for developing semi-explicit charge-preserved multi-symplectic schemes for stochastic cubic Schr\"odinger equations with multiplicative noise. Through numerical experiments that validate theoretical results, we demonstrate that the proposed stochastic semi-explicit symplectic scheme, which features a straightforward Newton iteration solver, outperforms the traditional stochastic midpoint scheme in terms of effectiveness and accuracy.