ORIGINAL ARTICLE
Newton's method for stochastic semilinear wave equations driven by multiplicative time-space noise
Henryk Leszczyński,
Monika Wrzosek,
Henryk Leszczyński
Institute of Mathematics, University of Gdańsk, Gdańsk, Poland
Search for more papers by this authorCorresponding Author
Monika Wrzosek
Institute of Mathematics, University of Gdańsk, Gdańsk, Poland
Correspondence
Monika Wrzosek, Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, Gdańsk 80-952, Poland.
Email: [email protected]
Search for more papers by this authorHenryk Leszczyński,
Monika Wrzosek,
Henryk Leszczyński
Institute of Mathematics, University of Gdańsk, Gdańsk, Poland
Search for more papers by this authorCorresponding Author
Monika Wrzosek
Institute of Mathematics, University of Gdańsk, Gdańsk, Poland
Correspondence
Monika Wrzosek, Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, Gdańsk 80-952, Poland.
Email: [email protected]
Search for more papers by this authorFirst published: 28 November 2022
Abstract
Semilinear wave equations with additive or one-dimensional noise are treatable by various iterative and numerical methods. We study more difficult models of semilinear wave equations with infinite dimensional multiplicative spatially correlated noise. Our proof of probabilistic second-order convergence of some iterative methods is based on Da Prato and Zabczyk's maximal inequalities.
REFERENCES
- 1R. Anton, D. Cohen, S. Larsson, and X. Wang, Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise, SIAM J. Numer. Anal. 54 (2016), 1093–1119.
- 2D. Cohen, S. Larsson, and M. Sigg, A trigonometric method for the linear stochastic wave equation, Siam J. Numer. Anal. 51 (2013), 204–222.
- 3D. Cohen and L. Quer-Sardanyons, A fully discrete approximation of the onedimensional stochastic wave equation, IMA J. Numer. Anal. 36 (2016), no. 1, 400–420.
- 4S. Cox, A. Jentzen, and F. Lindner, Weak convergence rates for temporal numerical approximations of stochastic wave equations with multiplicative noise, 2019, Preprint arXiv:1901.05535.
- 5J. Cui, J. Hong, L. Ji, and L. Sun, Strong convergence of a full discretization for stochastic wave equation with polynomial nonlinearity and addditive noise, 2019, Preprint arXiv:1909.00575.
- 6R. C. Dalang, The stochastic wave equation, a minicourse on stochastic partial differential equations, in: Lecture Notes in Math., 1962, Springer, Berlin, 2009, pp. 39–71.
- 7G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 1992.
10.1017/CBO9780511666223 Google Scholar
- 8K. J. Engel and R. Nagel, A short course on operator semigroups, Universitext, Springer, New York, 2006.
- 9O. Gonzalez and J. H. Maddocks, Extracting parameters for base-pair level models of DNA from molecular dynamics simulations, Theor. Chem. Acc. 106 (2001), 76–82.
- 10E. Hausenblas, Weak Approximation of the stochastic wave equation, J. Comput. Appl. Math. 235 (2010), 33–58.
- 11A. Hellemans, SOHO Probes Sun's interior by tuning in to its vibrations, Science 272 (1996), 1264–1265.
- 12A. Jentzen and P. E. Kloeden, Taylor approximations for stochastic partial differential equations. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 83, SIAM, Philadelphia, 2011.
10.1137/1.9781611972016 Google Scholar
- 13G. E. Karniadakis and Z. Zhang, Numerical methods for stochastic partial differential equations with white noise, Applied Mathematical Science, vol. 196, Springer, Berlin, 2017.
- 14S. Kawabata and T. Yamada, On Newton's method for stochastic differential equations, in: Séminaire de Probabilités XXV, Lecture Notes in Mathematics 1485, Springer, Berlin, 1991, pp. 121–137.
10.1007/BFb0100852 Google Scholar
- 15M. Kovács, S. Larsson, and F. Saedpanah, Finite element approximation of the linear stochastic wave equation with additive noise, SIAM J. Numer. Anal. 48 (2010), 408–427.
- 16R. Kruse, Strong and weak approximation of semilinear stochastic evolution equations. Lecture Notes in Mathematics, vol. 2093, Springer, Berlin, 2014.
10.1007/978-3-319-02231-4 Google Scholar
- 17Y. Li, Y. Wang, and W. Deng. Galerkin finite element approximations for stochastic space-time fractional wave equations. SIAM J. Numer. Anal. 55 (2017), no. 6, 3173–3202.
- 18H. Leszczyński and M. Wrzosek, Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion, Math. Biosci. Eng. 14 (2017), 237–248.
- 19H. Leszczyński and M. Wrzosek, Newton's method for nonlinear stochastic wave equations, Forum Math. 32 (2020), 595–605.
- 20H. Leszczyński, M. Matusik, and M. Wrzosek, Leap–frog method for stochastic functional wave equations, Electron. Trans. Numer. Anal. 52 (2020), 77–87.
- 21G. J. Lord, C. E. Powell, and T. Shardlow, An introduction to computational stochastic PDEs. Cambridge University Press, Cambridge, 2014.
10.1017/CBO9781139017329 Google Scholar
- 22A. Martin, S. M. Prigarin, and G. Winkler, Exact and fast numerical algorithms for the stochastic wave equation, Int. J. Comput. Math. 80 (2003), 1535–1541.
- 23L. J. de Naurois, A. Jentzen, and T. Welti, Lower bounds for weak approximation errors for spatial spectral Galerkin approximations of stochastic wave equations, in: A. Eberle, M. Grothaus, W. Hoh, M. Kassmann, W. Stannat, G. Trutnau (eds), Stochastic partial differential equations and related fields, vol. 229, Springer Proc. Math. Stat. Springer, Cham, 2018, pp. 237–248.
10.1007/978-3-319-74929-7_13 Google Scholar
- 24A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer, New York, 1983.
10.1007/978-1-4612-5561-1 Google Scholar
- 25C. Prévôt and M. Röckner, A concise course on stochastic partial differential equations, Lecture Notes in Mathematics, vol. 1905, Springer, Berlin, 2007.
- 26R. Qi and X. Wang. Error estimates of finite element method for semilinear stochastic strongly damped wave equation. IMA J. Numer. Anal. 39 (2019), no. 3, 1594–1626.
- 27L. Quer-Sardanyons and M. Sanz-Solé, Space semi-discretisations for a stochastic wave equation, Potential Anal. 24 (2006), 303–332.
- 28R. L. Schilling and L. Partzsch, Brownian motion, De Gruyter, Berlin, 2014.
10.1515/9783110307306 Google Scholar
- 29H. Schurz, A numerical method for nonlinear stochastic wave equations in , Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 14, Advances in Dynamical Systems, suppl. S2, 2007, 74–78.
- 30J. B. Walsh, An introduction to stochastic partial differential equations, in: École d'été de Probabilités de Saint-Flour XIV, Lecture Notes in Mathematics, vol. 1180, Springer, Berlin, 1986, pp. 265–439.
10.1007/BFb0074920 Google Scholar
- 31J. B. Walsh, On numerical solutions of the stochastic wave equation, Illinois J. Math. 50 (2006), 991–1018.
10.1215/ijm/1258059497 Google Scholar
- 32X. Wang, S. Gan, and J. Tang, Higher order strong approximations of semilinear stochastic wave equation with additive space-time white noise, SIAM J. Sci. Comput. 36 (2014), 2611–2632.
- 33M. Wrzosek, Newton's method for stochastic functional differential equations, Electron. J. Differential Equations 2012 (2012), 1–10.
- 34M. Wrzosek, Newton's method for stochastic functional evolution equations in Hilbert spaces, Mathematika 65 (2019), 542–556.
