Multi-symplectic Problems for Stochastic Hamiltonian System

Shanshan Jiang*, Jialin Hong**, Lijin Wang**

*Beijing University of Chemical Technology, Beijing , China

**Chinese Academy of Sciences, Beijing , China

Nanjing , Dec 15 , 2012

Stochastic Numerical Methods for Stochastic Korteweg-de Vries Equation

Stochastic Hamiltonian ODEs and Stochastic

Symplectic Structure

Stochastic Hamiltonian PDEs and Stochastic

Multi-Symplectic Conservation law

Outline:

Further Problems

Deterministic Hamiltonian ODEs have the form of

Proposition1[1]: The phase flows of the deterministic Hamiltonian ODEs preserve the symplectic structure:

Here, P and Q are d-dimensional variables.

Stochastic Hamiltonian ODEs are defined as

Here, P and Q are d-dimensional variables, and W(t) is the standard Wiener process, and o means Stratonovich product.

Proposition2[2]: The phase flow of the above system preserves the stochastic symplectic structure:

We have some conclusions:

1. The above two systems are called Hamiltonian systems, both deterministic and stochastic cases.

2. The above Hamiltonian systems possess some geometric property, i.e. the symplectic structures.

3. Many numerical methods are investigated to simulate these systems, especially those methods which can preserve the geometric structure.

Properties of various ODEs systems

SystemsDeterministic Hamiltonian

ODEs

Stochastic

Hamiltonian ODEs

ODE /SODE

Symplectic Structure

Symplectic Methods

Preservation Preservation

),(

) ,(

QPHQ

QPHP

p

q

)(),(),(

)(),() ,(

tdwQPGdtQPHdQ

tdwQPGdtQPHdP

pp

qq

dqdpdQdP dqdpdQdP

Deterministic Hamiltonian PDEs are written as

Proposition3[3]: The system possesses the multi-symplectic conservation law , which is the local geometric structure:

Here, M and K are skew-symmetric matrices.

are differential 2-form.

1. What kind of Stochastic Partial Differential Equations can be considered as the Stochastic Hamiltonian PDEs ?

2. Whether this kind of Stochastic Hamiltonian PDEs also possesses some kind of stochastic geometric properties ?

3. This kind of Stochastic Hamiltonian system is exist or not ? How about their practical significance of application ?

We ask some questions:

Properties of various PDEs systems

SystemsDeterministic

Hamiltonian PDEs

Stochastic

Hamiltonian PDEs

PDE/SPDE

?

Multi-symplectic Conservation law

?

Multi-symplectic Integrators

Preservation

?

)(zSKzMz xt

0)2

1()

2

1( KdzdzMdzdz xt

We propose a kind of Stochastic Hamiltonian PDEs:

is real-valued white noise, which is delta correlated in time, and either smooth or delta correlated in space.

Here, M and K are two skew-symmetric matrices.

There are some mathematical expression[4]:1. Define the cylindrical wiener process on , the space of square integrable functions associated to the stochastic basis

Theorem 1 [5]: The stochastic Hamiltonian PDE preserves the stochastic multi-symplectic conservation law locally in any definition domain :

2. is a sequence of independent real Brownian motions, is any orthonormal basis of 3. The space-time white noise has the form

Deterministic Korteweg-de Vries equation

Initial-boundary problem possesses infinite invariants functionals,

Introduce potential variable and momentum variable

Set with

The equation is transformed to the multi-symplectic PDE

Stochastic Korteweg–de Vries equation with additive noise:

Further set corresponding to the deterministic case.

represents the amplitude of noise source.

The equation is transformed to the stochastic multi-symplectic PDE:

The space-time white noise

Correlation function

Theorem 2: The stochastic Korteweg-de Vries equation preserves the stochastic multi-symplectic conservation law locally in any domain

Recursion of the average invariants,

We see that the global errors of the averages invariants are related to

Midpoint Rule Method (MP)

Numerical Methods:

Theorem 3: The discretization (MP) is a stochastic multi-symplectic integrator, and it can preserve the discrete multi-sysmplectic conservation law

Finally get 8-point MP Scheme:

Numerical Experiments

The profile of numerical solution as and

The profile of conservation laws as

The profile of conservation laws as

Ratio of transformation

We get some conclusions:

1. Korteweg-de Vries equation with additive noise can be considered as the Stochastic Hamiltonian PDE .

2. Stochastic Hamiltonian PDEs possesses some kind of stochastic geometric properties .

3. Multi-symplectic schemes can stably simulate the stochastic KdV equation for a long time interval, just as applied to the deterministic case.

1. The mean square orders of discrete integrators: theoretical proof and numerical simulations.

2. Various schemes, for example conservative schemes, for the stochastic Hamiltonian systems.

3. Other kind of partial differential equations which are included in the field of Stochastic Hamiltonian systems exist in practical significance of application.

Further Problems

[1] E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer-Verlag, 2002

[2] G. Milstein, M. Tretyakov, Stochastic Numierics for Mathematical Physics, Kluwer Axcademic Publisher, 1995

[3] T. Bridges, S.Reich, Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A, 284 (2001),184-193

[4] A. Debussche, J. Printems, Numerical Simulation of the Stochastic Korteweg-de Vries Equation, Phys. D, 134 (1999) 200-226

[5] S. Jiang, L. Wang, J. Hong, Stochastic Multi-symplectic Integrator for Stochastic Nonlinear Schrodinger Equation, Comm. Comput. Phys. (2013 accepted)

References:

Thanks for your attention!