Abstract
This paper aims to investigate the weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. We are interested in SPDEs where the nonlinear part is stronger than the linear part, also called stochastic reaction dominated transport equations. For such SPDEs, many standard numerical schemes lose their stability properties. Exponential Rosenbrock and Rosenbrock-type methods were proved to be efficient for such SPDEs, but only their strong convergence were recently analyzed. Here, we investigate the weak convergence of the Rosenbrock semi-implicit method. We obtain a weak convergence rate which is twice the rate of the strong convergence. Our error analysis does not rely on Malliavin calculus, but rather only uses the Kolmogorov equation and the smoothing properties of the resolvent operator resulting from the Rosenbrock semi-implicit approximation.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: 317210226
Funding statement: The work of J. D. Mukam was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 317210226 – SFB 1283.
Acknowledgements
The authors would like to thank the anonymous referees for their insightful comments that helped to improve this work.
References
[1] A. Andersson, M. Kovács and S. Larsson, Weak error analysis for semilinear stochastic Volterra equations with additive noise, J. Math. Anal. Appl. 437 (2016), no. 2, 1283–1304. 10.1016/j.jmaa.2015.09.016Suche in Google Scholar
[2] A. Andersson, R. Kruse and S. Larsson, Duality in refined Sobolev–Malliavin spaces and weak approximation of SPDE, Stoch. Partial Differ. Equ. Anal. Comput. 4 (2016), no. 1, 113–149. 10.1007/s40072-015-0065-7Suche in Google Scholar
[3] A. Andersson and S. Larsson, Weak convergence for a spatial approximation of the nonlinear stochastic heat equation, Math. Comp. 85 (2016), no. 299, 1335–1358. 10.1090/mcom/3016Suche in Google Scholar
[4] A. Andersson and F. Lindner, Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise, Discrete Contin. Dyn. Syst. Ser. B 24 (2019), no. 8, 4271–4294. 10.3934/dcdsb.2019081Suche in Google Scholar
[5] P. R. Beesack, More generalised discrete Gronwall inequalities, ZAMM Z. Angew. Math. Mech. 65 (1985), no. 12, 583–595. 10.1002/zamm.19850651202Suche in Google Scholar
[6] C.-E. Bréhier and A. Debussche, Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient, J. Math. Pures Appl. (9) 119 (2018), 193–254. 10.1016/j.matpur.2018.08.010Suche in Google Scholar
[7] P.-L. Chow, Stochastic Partial Differential Equations, Chapman & Hall/CRC Appl. Math. Nonlinear Sci. Ser., Chapman & Hall/CRC, Boca Raton, 2007. Suche in Google Scholar
[8] J. Cui and J. Hong, Strong and weak convergence rates of a spatial approximation for stochastic partial differential equation with one-sided Lipschitz coefficient, SIAM J. Numer. Anal. 57 (2019), no. 4, 1815–1841. 10.1137/18M1215554Suche in Google Scholar
[9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl. 152, Cambridge University, Cambridge, 2014. 10.1017/CBO9781107295513Suche in Google Scholar
[10] A. Debussche, Weak approximation of stochastic partial differential equations: The nonlinear case, Math. Comp. 80 (2011), no. 273, 89–117. 10.1090/S0025-5718-2010-02395-6Suche in Google Scholar
[11] A. Debussche and J. Printems, Weak order for the discretization of the stochastic heat equation, Math. Comp. 78 (2009), no. 266, 845–863. 10.1090/S0025-5718-08-02184-4Suche in Google Scholar
[12] H. Fujita and T. Suzuki, Evolution problems, Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam (1991), 789–928. 10.1016/S1570-8659(05)80043-2Suche in Google Scholar
[13] M. Geissert, M. Kovács and S. Larsson, Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise, BIT 49 (2009), no. 2, 343–356. 10.1007/s10543-009-0227-ySuche in Google Scholar
[14] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer, Berlin, 1981. 10.1007/BFb0089647Suche in Google Scholar
[15] A. Jentzen, P. Kloeden and G. Winkel, Efficient simulation of nonlinear parabolic SPDEs with additive noise, Ann. Appl. Probab. 21 (2011), no. 3, 908–950. 10.1214/10-AAP711Suche in Google Scholar
[16] M. Kovács, S. Larsson and F. Lindgren, Strong convergence of the finite element method with truncated noise for semilinear parabolic stochastic equations with additive noise, Numer. Algorithms 53 (2010), no. 2–3, 309–320. 10.1007/s11075-009-9281-4Suche in Google Scholar
[17] R. Kruse, Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise, IMA J. Numer. Anal. 34 (2014), no. 1, 217–251. 10.1093/imanum/drs055Suche in Google Scholar
[18] R. Kruse and M. Scheutzow, A discrete stochastic Gronwall lemma, Math. Comput. Simulation 143 (2018), 149–157. 10.1016/j.matcom.2016.07.002Suche in Google Scholar
[19] S. Larsson, Nonsmooth data error estimates with applications to the study of the long-time behavior of finite element solutions of semilinear parabolic problems, Preprint 1992-36, Department of Mathematics, Chalmers University of Technology, 1992. Suche in Google Scholar
[20] G. J. Lord and A. Tambue, Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative & additive noise, IMA J. Numer. Anal. 2 (2013), 515–543. 10.1093/imanum/drr059Suche in Google Scholar
[21] J. D. Mukam, Some numerical techniques for approximating semilinear parabolic (stochastic) partial differential equations, PhD Thesis, Fakultät für Mathematik, Technische Universität Chemnitz, 2021, https://nbn-resolving.org/urn:nbn:de:bsz:ch1-qucosa2-761232. Suche in Google Scholar
[22] J. D. Mukam and A. Tambue, A note on exponential Rosenbrock–Euler method for the finite element discretization of a semilinear parabolic partial differential equation, Comput. Math. Appl. 76 (2018), no. 7, 1719–1738. 10.1016/j.camwa.2018.07.025Suche in Google Scholar
[23] J. D. Mukam and A. Tambue, Strong convergence analysis of the stochastic exponential Rosenbrock scheme for the finite element discretization of semilinear SPDEs driven by multiplicative and additive noise, J. Sci. Comput. 74 (2018), no. 2, 937–978. 10.1007/s10915-017-0475-ySuche in Google Scholar
[24] J. D. Mukam and A. Tambue, Strong convergence of a stochastic Rosenbrock-type scheme for the finite element discretization of semilinear SPDEs driven by multiplicative and additive noise, Stochastic Process. Appl. 130 (2020), no. 8, 4968–5005. 10.1016/j.spa.2020.02.008Suche in Google Scholar
[25] T. Nambu, Characterization of the domain of fractional powers of a class of elliptic differential operators with feedback boundary conditions, J. Differential Equations 136 (1997), no. 2, 294–324. 10.1006/jdeq.1996.3252Suche in Google Scholar
[26] C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Math. 1905, Springer, Berlin, 2007. Suche in Google Scholar
[27] A. Tambue, Efficient numerical schemes for porous media flow, PhD Thesis, Department of Mathematics, Heriot–Watt University, 2010. Suche in Google Scholar
[28] A. Tambue, I. Berre and J. M. Nordbotten, Efficient simulation of geothermal processes in heteregeneous porous media based on the exponential Rosenbrock–Euler and Rosenbrock-type methods, Adv. Water Resour. 53 (2013), 250–262. 10.1016/j.advwatres.2012.12.004Suche in Google Scholar
[29] A. Tambue and J. D. Mukam, Weak convergence of the finite element method for semilinear parabolic SPDEs driven by additive noise, Results Appl. Math. 17 (2023), Paper No. 100351. 10.1016/j.rinam.2022.100351Suche in Google Scholar
[30] A. Tambue and J. M. T. Ngnotchouye, Weak convergence for a stochastic exponential integrator and finite element discretization of stochastic partial differential equation with multiplicative & additive noise, Appl. Numer. Math. 108 (2016), 57–86. 10.1016/j.apnum.2016.04.013Suche in Google Scholar
[31] X. Wang, Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus, Discrete Contin. Dyn. Syst. 36 (2016), no. 1, 481–497. 10.3934/dcds.2016.36.481Suche in Google Scholar
[32] X. Wang, Strong convergence rates of the linear implicit Euler method for the finite element discretization of SPDEs with additive noise, IMA J. Numer. Anal. 37 (2017), no. 2, 965–984. 10.1093/imanum/drw016Suche in Google Scholar
[33] X. Wang and S. Gan, Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise, J. Math. Anal. Appl. 398 (2013), no. 1, 151–169. 10.1016/j.jmaa.2012.08.038Suche in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
