×

High-accuracy time discretization of stochastic fractional diffusion equation. (English) Zbl 07435363

Summary: A high-accuracy time discretization is discussed to numerically solve the nonlinear fractional diffusion equation forced by a space-time white noise. The main purpose of this paper is to improve the temporal convergence rate by modifying the semi-implicit Euler scheme. The solution of the equation is only Hölder continuous in time, which is disadvantageous to improve the temporal convergence rate. Firstly, the system is transformed into an equivalent form having better regularity than the original one in time. But the regularity of nonlinear term remains unchanged. Then, combining Lagrange mean value theorem and independent increments of Brownian motion leads to a higher accuracy discretization of nonlinear term which ensures the implementation of the proposed time discretization scheme without loss of convergence rate. Our scheme can improve the convergence rate from \(\min \{\frac{\gamma}{2\alpha},\frac{1}{2}\}\) to \(\min \{\frac{\gamma}{\alpha},1\}\) in the sense of mean-squared \(L^2\)-norm. The theoretical error estimates are confirmed by extensive numerical experiments.

MSC:

65Mxx Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
26A33 Fractional derivatives and integrals
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65L20 Stability and convergence of numerical methods for ordinary differential equations
65C30 Numerical solutions to stochastic differential and integral equations
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Blömker, D.; Kamrani, M., Numerically computable a posteriori-bounds for the stochastic Allen-Cahn equation, Bit. Numer. Math., 59, 647-673 (2019) · Zbl 1433.60073
[2] Bou-Rabee, N., SPECTRWM: spectral random walk method for the numerical solution of stochastic partial differential equations, SIAM Rev., 60, 2, 386-406 (2018) · Zbl 1407.60084
[3] Bréhier, CE, Approximation of the invariant measure with an Euler scheme for stochastic PDEs driven by space-time white noise, Potential Anal., 40, 1-40 (2014) · Zbl 1286.35010
[4] Chen, CC; Hong, JL; Ji, L., Mean-square convergence of a semidiscrete sheme for stochastic maxwell equations, SIAM J. Numer. Anal., 57, 2, 728-750 (2019) · Zbl 1422.60112
[5] Cui, JB; Hong, JL; Liu, ZH; Zhou, W., Strong convergence rate of splitting schemes for stochastic nonlinear Schrödinger equations, J. Differ. Equ., 266, 9, 5625-5663 (2019) · Zbl 1431.60066
[6] Deng, WH; Hou, R.; Wang, WL; Xu, PB, Modeling Anomalous Diffusion: From Statistics and Mathematics (2020), Singapore: World Scientific, Singapore · Zbl 1453.35001
[7] Dybiec, B.; Kleczkowski, A.; Gilligan, CA, Modelling control of epidemics spreading by long-range interactions, J. R. Soc. Interface., 6, 39, 941-950 (2009)
[8] Choi, JH; Han, BS, A regularity theory for stochastic partial differential equations with a super-linear diffusion coefficient and a spatially homogeneous colored noise, Stochastic Process Appl., 135, 1-30 (2021) · Zbl 1471.60097
[9] Chow, P., Stochastic Partial Differential Equations (2007), New York: Chapman & Hall/CRC, New York · Zbl 1134.60043
[10] Gunzburger, M.; Li, BY; Wang, JL, Sharp convergence rates of time discretization for stochastic time-fractional PDEs subject to additive space-time white noise, Math. Comput., 88, 318, 1715-1741 (2019) · Zbl 1418.60077
[11] Hong, JL; Wang, X.; Zhang, LY, Parareal exponential \(\theta \)-scheme for longtime simulation of stochastic Schrödinger equations with weak damping, SIAM J. Sci. Comput., 41, 6, B1155-B1177 (2019) · Zbl 1433.60076
[12] Hu, Y.; Huang, J.; Lê, K.; Nualart, D.; Tindel, S., Stochastic heat equation with rough dependence in space, Ann. Probab., 45, 6, 4561-4616 (2017) · Zbl 1393.60066
[13] Laptev, A., Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces, J. Funct. Anal., 151, 2, 531-545 (1997) · Zbl 0892.35115
[14] Li, P.; Yau, ST, On the Schrödinger equation and the eigenvalue problem, Comm Math Phys., 88, 309-318 (1983) · Zbl 0554.35029
[15] Liu, X.; Deng, WH, Numerical approximation for fractional diffusion equation forced by a tempered fractional Gaussian noise, J. Sci. Comput., 84, 1, 1-28 (2020) · Zbl 1434.65267
[16] Liu, X.; Deng, WH, Higher order approximation for stochastic space fractional wave equation forced by an additive space-time Gaussian noise, J. Sci. Comput., 87, 1, 1-29 (2021) · Zbl 1477.65176
[17] Liu, ZH; Qiao, ZH, Strong approximation of monotone stochastic partial differential equations driven by white noise, IMA J. Numer. Anal., 40, 2, 1074-1093 (2020) · Zbl 1464.65148
[18] Mörters, P.; Peres, Y., Brownian Motion (2010), Cambridge: Cambridge University Press, Cambridge · Zbl 1243.60002
[19] van Neerven, JMAM; Veraar, MC; Weis, L., Stochastic integration in UMD Banach spaces, Ann. Probab., 35, 4, 1438-1478 (2007) · Zbl 1121.60060
[20] Nochetto, RH; Otárola, E.; Salgado, AJ, A PDE approach to fractional diffusion in general domains: a priori error analysis, Found. Comput Math., 15, 733-791 (2015) · Zbl 1347.65178
[21] Prato, GD; Zabczyk, J., Stochastic Equations in Infinite Dimensions (2014), Cambridge: Cambridge University Press, Cambridge · Zbl 1317.60077
[22] Song, R., Vondrac̆ek, Z,: Potential theory of subordinate killed Brownian motion in a domain. Probab. Theory Relat. Fields 125, 578-592 (2003) · Zbl 1022.60078
[23] Song, J.; Song, X.; Zhang, Q., Nonlinear Feynman-Kac formulas for stochastic partial differential equations with space-time noise, SIAM J. Math. Anal., 51, 2, 955-990 (2019) · Zbl 1411.60101
[24] Strauss, WA, Partial Differential Equations: An Introduction (2008), New York: Wiley, New York · Zbl 1160.35002
[25] Thomée, V., Galerkin Finite Element Methods for Parabolic Problems (2006), Berlin: Springer, Berlin · Zbl 1105.65102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.