Discrete gradient methods and linear projection methods for preserving a conserved quantity of stochastic differential equations

ABSTRACT

Numerical methods preserving a conserved quantity for stochastic differential equations are considered. A class of discrete gradient methods based on the skew-gradient form is constructed, and the sufficient condition of convergence order 1 in the mean-square sense is given. Then a class of linear projection methods is constructed. The relationship of the two classes of methods for preserving a conserved quantity is proved, which is, the constructed linear projection methods can be considered as a subset of the constructed discrete gradient methods. Numerical experiments verify our theory and show the efficiency of proposed numerical methods.

Keywords:

• Stochastic differential equations
• conserved quantity
• mean-square convergence
• discrete gradient methods
• linear projection methods

AMS Classification:

• 60H10
• 37N30
• 65P10

Acknowledgments

The authors would like to express their appreciation to the anonymous referees for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (Nos. 11501150 and 11401140), the Key Project of Science and Technology of Weihai (No. 2014DXGJ14), Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (No. HIT. NSRIF. 2016081), and the Scientific Research Foundation of Harbin Institute of Technology at Weihai (No. HIT(WH)201319).