Shanshan Jiang*, Jialin Hong**, Lijin Wang**
*Beijing University of Chemical Technology, Beijing , Chi na
**Chinese Academy of Sciences, Beijing , Chi na
Nanjing , Dec 15 , 2012



 Stochastic Numerical Methods for Stochastic
Korteweg-de Vries Equation


Deterministic Hamiltonian ODEs have the form of
Here, P and Q are d-dimensional variables.
Proposition1[1]: The phase flows of the deterministic
Hamiltonian
ODEs preserve the symplectic structure :

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 :

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.

Systems
ODE
/SODE
Deterministic
Hamiltonian
ODEs
 
H q
( P , Q )

H p
( P , Q )
Stochastic
Hamiltonian ODEs dP
 
H q
( P , Q ) dt

G q
( P , Q )  dw ( t ) dQ

H p
( P , Q ) dt

G p
( P , Q )  dw ( t )
Symplectic
Structure dP
 dQ
 dp
 dq
Symplectic
Methods
Preservation dP
 dQ
 dp
 dq
Preservation

Deterministic Hamiltonian PDEs are written as
Here, M and K are skew-symmetric matrices.
Proposition3[3]: The system possesses the multi-symplectic conservation law , which is the local geometric structure: 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 ?

Systems
Deterministic
Hamiltonian PDEs
Stochastic
Hamiltonian
PDEs
PDE/SPDE
Mz t

Kz x
 
S ( z )
?
Multisymplectic
Conservation law
 t
(
1
2 dz

Mdz )
  x
(
1
2 dz

Kdz )

0
Multisymplectic
Integrators
Preservation
?
?

Here, M and K are two skew-symmetric matrices.
is real-valued white noise, which is delta correlated in time, and either smooth or delta correlated in space.
There are some mathematical expression[4]:
1.
Define the cylindrical wiener process on , the space of square integrable functions associated to the stochastic basis

2 . is a sequence of independent real Brownian motions, is any orthonormal basis of
3. The space-time white noise has the form
Theorem 1 [5]: The stochastic Hamiltonian PDE preserves the stochastic multi-symplectic conservation law locally in any definition domain :

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

Numerical Methods : Midpoint Rule Method (MP)
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

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.

[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)