An Accelerated Convergence Scheme for Solving Stochastic Fractional Diffusion Equation

doi:10.1007/s42967-023-00342-1

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (4): 1444-1461.doi: 10.1007/s42967-023-00342-1

• ORIGINAL PAPERS • 上一篇下一篇

An Accelerated Convergence Scheme for Solving Stochastic Fractional Diffusion Equation

Xing Liu

School of Mathematics and Statistics, Hubei Normal University, Huangshi, 430205, Hubei, China

收稿日期:2023-04-18 修回日期:2023-08-16 接受日期:2023-10-17 出版日期:2024-01-16 发布日期:2024-01-16
通讯作者: Xing Liu,E-mail:2718826413@qq.com E-mail:2718826413@qq.com
基金资助:
Not applicable.

An Accelerated Convergence Scheme for Solving Stochastic Fractional Diffusion Equation

Xing Liu

School of Mathematics and Statistics, Hubei Normal University, Huangshi, 430205, Hubei, China

Received:2023-04-18 Revised:2023-08-16 Accepted:2023-10-17 Online:2024-01-16 Published:2024-01-16
Supported by:
Not applicable.

摘要/Abstract

摘要： An accelerated convergence scheme for temporal approximation of stochastic partial differential equation is presented. First, the regularity of the mild solution is provided. Combining the Itô formula and the remainder term of the exponential Euler scheme, this paper proposes a high accuracy time discretization method. Based on regularity results, a strong convergence rate for the discretization error $O\left(\tau^{\frac{3}{2}-\epsilon}\right)$ is proved for arbitrarily small $\epsilon$＞0. Here $\tau$ is the uniform time step size. Finally, the theoretical results are verified by several numerical experiments.

关键词: Accelerated convergence scheme, Temporal approximation, It? formula, Remainder term

Abstract: An accelerated convergence scheme for temporal approximation of stochastic partial differential equation is presented. First, the regularity of the mild solution is provided. Combining the Itô formula and the remainder term of the exponential Euler scheme, this paper proposes a high accuracy time discretization method. Based on regularity results, a strong convergence rate for the discretization error $O\left(\tau^{\frac{3}{2}-\epsilon}\right)$ is proved for arbitrarily small $\epsilon$＞0. Here $\tau$ is the uniform time step size. Finally, the theoretical results are verified by several numerical experiments.

Key words: Accelerated convergence scheme, Temporal approximation, It? formula, Remainder term

中图分类号:

Xing Liu. An Accelerated Convergence Scheme for Solving Stochastic Fractional Diffusion Equation[J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1444-1461.

参考文献

[1] Bréhier, C.E.: Approximation of the invariant measure with an Euler scheme for stochastic PDEs driven by space-time white noise. Potential Anal. 40, 1-40 (2014). https://doi.org/10.1007/s11118-013-9338-9
[2] Bréhier, C.E., Cui, J., Hong, J.: Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen-Cahn equation. IMA J. Numer. Anal. 39, 2096-2134 (2019). https://doi.org/10.1093/imanum/dry052
[3] Bréhier, C.E., Kopec, M.: Approximation of the invariant law of SPDEs: error analysis using a Poisson equation for a full-discretization scheme. IMA J. Numer. Anal. 37, 1375-1410 (2017). https://doi.org/10.1093/imanum/drw030
[4] Bréhier, C.E., Vilmart, G.: High order integrator for sampling the invariant distribution of a class of parabolic stochastic PDEs with additive space-time noise. SIAM J. Sci. Comput. 38, 2283-2306 (2016). https://doi.org/10.1137/15M1021088
[5] Cui, J., Hong, J.: Strong and weak convergence rates of a spatial approximation for stochastic partial differential equation with one-sided Lipschitz coefficient. SIAM J. Numer. Anal. 57, 1815-1841 (2019). https://doi.org/10.1137/18M1215554
[6] Cui, J., Hong, J., Sun, L.: Weak convergence and invariant measure of a full discretization for parabolic SPDEs with non-globally Lipschitz coefficients. Stoch. Process. Appl. 134, 55-93 (2021). https://doi.org/10.1016/j.spa.2020.12.003
[7] Cusimano, N., del Teso, F., Gerardo-Giorda, L., Pagnini, G.: Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions. SIAM J. Numer. Anal. 56, 1243-1272 (2018). https://doi.org/10.1137/17M1128010
[8] Geissert, M., Kovács, M., Larsson, S.: Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise. BIT 49, 343-356 (2009). https://doi.org/10.1007/s10543-009-0227-y
[9] Grebenkov, D.S., Nguyen, B.T.: Geometrical structure of Laplacian eigenfunctions. SIAM Rev. 55, 601-667 (2013). https://doi.org/10.1137/120880173
[10] Jentzen, A.: Higher order pathwise numerical approximations of SPDEs with additive noise. SIAM J. Numer. Anal. 49, 642-667 (2011). https://doi.org/10.1137/080740714
[11] Kruse, R., Wu, Y.: A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations. Math. Comp. 88, 2793-2825 (2019). https://doi.org/10.1090/mcom/3421
[12] Liu, X.: High-accuracy time discretization of stochastic fractional diffusion equation. J. Sci. Comput. 90, 1-24 (2022). https://doi.org/10.1007/s10915-021-01710-w
[13] Liu, X.: Strong approximation for fractional wave equation forced by fractional Brownian motion with Hurst parameter H∈(0, 1/2). J. Comput. Appl. Math. 432, 115285 (2023). https://doi.org/10.1016/j.cam.2023.115285
[14] Liu, X., Deng, W.H.: Numerical approximation for fractional diffusion equation forced by a tempered fractional Gaussian noise. J. Sci. Comput. 84, 1-28 (2020). https://doi.org/10.1007/s10915-020-01271-4
[15] Liu, X., Deng, W.H.: Higher order approximation for stochastic space fractional wave equation forced by an additive space-time Gaussian noise. J. Sci. Comput. 87, 1-29 (2021). https://doi.org/10.1007/s10915-021-01415-0
[16] Liu, Z., Qiao, Z.: Strong approximation of monotone stochastic partial differential equations driven by white noise. IAM J. Numer. Anal. 40, 1074-1093 (2020). https://doi.org/10.1093/imanum/dry088
[17] Nochetto, R.H., Otárola, E., Salgado, A.J.: A PDE approach to fractional diffusion in general domains: a priori error analysis. Comput. Math. 15, 733-791 (2015). https://doi.org/10.1007/s10208-014-9208-x
[18] Song, R., Vondracek, Z.: Potential theory of subordinate killed Brownian motion in a domain. Probab. Theory Relat. Fields. 125, 578-592 (2003). https://doi.org/10.1007/s00440-002-0251-1
[19] Tambue, A., Ngnotchouye, J.M.T.: Weak convergence for a stochastic exponential integrator and finite element discretization of stochastic partial differential equation with multiplicative & additive noise. Appl. Numer. Math. 108, 57-86 (2016). https://doi.org/10.1016/j.apnum.2016.04.013
[20] Wang, X.: Strong convergence rates of the linear implicit Euler method for the finite element discretization of SPDEs with additive noise. IAM J. Numer. Anal. 37, 965-984 (2017). https://doi.org/10.1093/imanum/drw016
[21] Wang, X.: An efficient explicit full-discrete scheme for strong approximation of stochastic Allen-Cahn equation. Stoch. Process. Appl. 130, 6271-6299 (2020). https://doi.org/10.1016/j.spa.2020.05.011
[22] Wang, X., Gan, S.: Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise. J. Math. Anal. Appl. 398, 151-169 (2013). https://doi.org/10.1016/j.jmaa.2012.08.038

An Accelerated Convergence Scheme for Solving Stochastic Fractional Diffusion Equation

An Accelerated Convergence Scheme for Solving Stochastic Fractional Diffusion Equation

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

编辑推荐

Metrics

本文评价

[1]	Xuan Zhao, Zhongqin Xue. Efficient Variable Steps BDF2 Scheme for the Two-Dimensional Space Fractional Cahn-Hilliard Model[J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1489-1515.
[2]	Dildar Ahmad, Amjad Ali, Kamal Shah, Bahaaeldin Abdalla, Thabet Abdeljawad. On Nonlinear Analysis for Multi-term Delay Fractional Differential Equations Under Hilfer Derivative[J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1516-1539.
[3]	Baoli Yin, Yang Liu, Hong Li. Sharp Error Analysis for Averaging Crank-Nicolson Schemes with Corrections for Subdiffusion with Nonsmooth Solutions[J]. Communications on Applied Mathematics and Computation, 2025, 7(3): 865-884.
[4]	Tarek Aboelenen. Efficient and Accurate Spectral Method for Solving Fractional Differential Equations on the Half Line Using Orthogonal Generalized Rational Jacobi Functions[J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1419-1443.
[5]	Divyansh Pandey, Prashant K. Pandey, Rajesh K. Pandey. Variational and Numerical Approximations for Higher Order Fractional Sturm-Liouville Problems[J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1398-1418.
[6]	Qiaoqiao Dai, Dongxia Li. The L2-1_σ/LDG Method for the Caputo Diffusion Equation with a Variable Coefficient[J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1378-1397.
[7]	Yuri Dimitrov. Approximations of the Fractional Integral and Numerical Solutions of Fractional Integral Equations[J]. Communications on Applied Mathematics and Computation, 2021, 3(3): 545-569.
[8]	K. A. Mustapha, K. M. Furati, O. M. Knio, O. P. Le Maître. A Finite Diference Method for Space Fractional Diferential Equations with Variable Difusivity Coefcient[J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 671-688.
[9]	Junying Cao, Ziqiang Wang, Chuanju Xu. A High-Order Scheme for Fractional Ordinary Diferential Equations with the Caputo-Fabrizio Derivative[J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 179-199.
[10]	Xue-lei Lin, Pin Lyu, Michael K. Ng, Hai-Wei Sun, Seakweng Vong. An Efcient Second-Order Convergent Scheme for One-Side Space Fractional Difusion Equations with Variable Coefcients[J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 215-239.
[11]	Can Li, Shuming Liu. Local Discontinuous Galerkin Scheme for Space Fractional Allen-Cahn Equation[J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 73-91.
[12]	Yubo Yang, Fanhai Zeng. Numerical Analysis of Linear and Nonlinear Time-Fractional Subdiffusion Equations[J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 621-637.
[13]	Changpin Li, Dongxia Li, Zhen Wang. L1/LDG Method for the Generalized Time-Fractional Burgers Equation in Two Spatial Dimensions[J]. Communications on Applied Mathematics and Computation, 2023, 5(4): 1299-1322.
[14]	Zhengnan Dong, Enyu Fan, Ao Shen, Yuhao Su. Three Kinds of Discrete Formulae for the Caputo Fractional Derivative[J]. Communications on Applied Mathematics and Computation, 2023, 5(4): 1446-1468.
[15]	Enyu Fan, Jingshu Wu, Shaoying Zeng. On the Fractional Derivatives with an Exponential Kernel[J]. Communications on Applied Mathematics and Computation, 2023, 5(4): 1655-1673.