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Novel semi-explicit symplectic schemes for nonseparable stochastic Hamiltonian systems

Jialin Hong, Baohui Hou, Liying Sun

TL;DR

The paper tackles preserving stochastic geometric structure and invariants for nonseparable SHSs and for stochastic cubic Schrödinger equations with multiplicative noise. It introduces augmented Hamiltonians and symmetric projection to build a two-copy extended SHS and then derives semi-explicit SES-SP schemes that are symplectic almost surely in the original space and can preserve a quadratic invariant $Q_\kappa(Z)=\tfrac12 Z^T \kappa Z$ when $\gamma_i=0$. For SPDEs, it develops semi-explicit multi-symplectic discretizations that preserve the discrete multisymplectic law and charge conservation, using a splitting that yields explicit solvable subsystems. Numerical results demonstrate improved efficiency and reliable invariant preservation compared with the stochastic midpoint scheme, validating the practicality of the proposed methods for long-time simulations in physics and engineering.

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 Schrödinger 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.

Novel semi-explicit symplectic schemes for nonseparable stochastic Hamiltonian systems

TL;DR

The paper tackles preserving stochastic geometric structure and invariants for nonseparable SHSs and for stochastic cubic Schrödinger equations with multiplicative noise. It introduces augmented Hamiltonians and symmetric projection to build a two-copy extended SHS and then derives semi-explicit SES-SP schemes that are symplectic almost surely in the original space and can preserve a quadratic invariant when . For SPDEs, it develops semi-explicit multi-symplectic discretizations that preserve the discrete multisymplectic law and charge conservation, using a splitting that yields explicit solvable subsystems. Numerical results demonstrate improved efficiency and reliable invariant preservation compared with the stochastic midpoint scheme, validating the practicality of the proposed methods for long-time simulations in physics and engineering.

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 Schrödinger 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.
Paper Structure (3 sections, 4 theorems, 75 equations, 10 figures, 4 tables)

This paper contains 3 sections, 4 theorems, 75 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Stochastic semi-explicit symplectic schemes for SHS
  3. Stochastic semi-explicit multi-symplectic schemes for stochastic cubic Schrödinger equations with multiplicative noise

Key Result

Theorem 2.1

If the extended phase space scheme based on $\mathcal{F}_{\Delta t}$ is symplectic, then the stochastic semi-explicit scheme based on $\widetilde{\mathcal{F}}_{\Delta t}$ preserves symplectic structure almost surely, i.e., where $n\in\{0,1,\ldots,N-1\}$.

Figures (10)

  • Figure 1: Time evolutions of norm $\|(X-U, Y-V)\|$ with $\Delta t = 1\times 10^{-2}, c = 0.5, \gamma = 1$: (left)SES-SP-1, (right) SES-SP-2.
  • Figure 2: Mean-square convergence order of various schemes in time with $T=1, c = 0.15, \gamma = 0.01$.
  • Figure 3: Relative Hamiltonian $H_0$ of various schemes: with $T =100, \Delta t = 1\times 10^{-4}, c = 0.1, \gamma = 0$: (1)SES-SP-1, (2) SES-SP-2, (3)Midpoint, (4)Symplectic Euler.
  • Figure 4: Mean-square convergence order of various schemes with $T=1, c = 0.5, \gamma = 0.5$ and $\Delta t=2^{-s}, s\in\{5, 6, 7, 8\}$.
  • Figure 5: Relative Casimir function with $T =100, \Delta t = 1.25\times 10^{-2}, c = 1, \gamma = 1$: (1)SES-SP-1, (2) SES-SP-2, (3)Midpoint.
  • ...and 5 more figures

Theorems & Definitions (14)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Remark 3.1
  • Remark 3.2
  • ...and 4 more