Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods
Abstract
:1. Introduction
2. Preliminaries
- Denoteas the trees with only one vertex of color l,,
- Denoteas the tree formed by joining subtreeseach by a single branch to a common root.
- ,
- ,
- If, then.
3. Improved ESRK Methods
Algorithm 1 Improved ESRK method |
Step 1. Set the initial value and choose a constant . |
Step 2. |
if then |
end the procedure; |
else |
compute by the ESRK method; |
end if |
Step 3. |
if then |
let and go to Step 2; |
else |
go to Step 4; |
end if |
Step 4. Compute such that reaches the minimum. Compute
|
4. Numerical Examples
4.1. Stochastic Kubo Oscillator
- In order to show the proposed methods have the same mean-square convergence order as the original method. The mean-square convergence rate of methods (18) and (30) are displayed in Figure 1. Here, the mean-square errors are computed over 1000 simple paths with five different step sizes: , , , , and .
- In order to illustrate the advantages of the proposed methods in computational efficiency, we also apply the stochastic projection method and the stochastic discrete gradient method to solve (38) with . Figure 2 displays the mean-square sample errors and the corresponding CPU time of the stochastic projection method, the stochastic discrete gradient method, the Platen method, method (18), and method (30).
- In order to show the proposed methods’ ability to preserve the conserved quantity and get more accurate numerical solutions than the original method. We display the numerical sample paths and absolute errors in conserved quantity I of the Platen method, method (18), and method (30) in Figure 3 and Figure 4, respectively. Here, .
4.2. Stochastic Mathematical Pendulum
4.3. Stochastic Rigid Body
4.4. Stochastic Kepler Problem
5. Conclusions and Discussion
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Wang, Z.; Ma, Q.; Ding, X. Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods. Mathematics 2020, 8, 2195. https://doi.org/10.3390/math8122195
Wang Z, Ma Q, Ding X. Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods. Mathematics. 2020; 8(12):2195. https://doi.org/10.3390/math8122195
Chicago/Turabian StyleWang, Zhenyu, Qiang Ma, and Xiaohua Ding. 2020. "Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods" Mathematics 8, no. 12: 2195. https://doi.org/10.3390/math8122195