An Accelerated Convergence Scheme for Solving Stochastic Fractional Diffusion Equation

doi:10.1007/s42967-023-00342-1

Abstract

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

CLC Number:

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

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References

[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