Skip to main content Accessibility help
×
Home

Difference Approximation of Stochastic Elastic Equation Driven by Infinite Dimensional Noise

  • Yinghan Zhang (a1), Xiaoyuan Yang (a1) and Ruisheng Qi (a1)

Abstract

An explicit difference scheme is described, analyzed and tested for numerically approximating stochastic elastic equation driven by infinite dimensional noise. The noise processes are approximated by piecewise constant random processes and the integral formula of the stochastic elastic equation is approximated by a truncated series. Error analysis of the numerical method yields estimate of convergence rate. The rate of convergence is demonstrated with numerical experiments.

Copyright

Corresponding author

*Corresponding author. Email address: zhangyinghan007@126.com (Y.-H. Zhang)

References

Hide All
[1]Allen, E.J., Novosel, S.J., and Zhang, Z., Finite element and difference approximation of some linear stochastic partical differential equations, Stoch. Stoch. Rep., vol. 64 (1998), pp. 117142.
[2]Bally, V. and Talay, D., The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function, Probab. Theory Relat. Field., vol. 104 (1996), pp. 4360.
[3]Brzeźniak, Z., Maslowski, B. and Seidler, J., Stochastic nonlinear beam equations, Probab. Theory Relat. Field., vol. 132 (2005), pp. 119149.
[4]Cao, Y.Z., Yang, H.T. and Yin, L., Finite element methods for semilinear elliptic stochastic partial differential equations, Numer. Math., vol. 106 (2007), pp. 181198.
[5]Cao, Y.Z., Zhang, R. and Zhang, K., Finite element and discontinuous Galerkin method for stochastic Helmholtz equation in two and three dimensions, J. Comput. Math., vol. 5 (2008), pp. 702715.
[6]Chow, P.L. and Menaldi, J.L., Stochastic PDE for nonlinear Vibration of elastic panels, Diff. Int. Eqns., vol. 12 (1999), pp. 419434.
[7]Debussche, A. and Printems, J., Weak order for the discretization of the stochastic heat equation, Math. Comput., vol. 78 (2009), pp. 845863.
[8]Du, Q. and Zhang, T.Y., Numerical approximation of some linear stochastic partial differential equations driven by special additive noise, SIAM J. Numer. Anal., vol. 4 (2002), pp. 14211445.
[9]Etheridge, A., Stochastic Partial Differential Equations, Cambridge University Press, Cambridge, 1995.
[10]Feng, J.F., Lei, G.Y. and Qian, M.P., Second-order methods for solving stochastic differential equations, J. Comput. Math., vol. 10 (1992), pp. 376387.
[11]Galal, O.H., El-Tawil, M.A. and Mahmoud, A.A., Stochastic beam equations under random dynamic loads, Inter. J. Solid. Struct., vol. 39 (2002), pp. 10311040.
[12]Gard, T.C., Intruduction to Stochastic Differential Equations, Marcel Decker, New York, 1988.
[13]Gyöngy, I. and Nualart, D., Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise, Potential Anal., vol. 7 (1997), pp. 725757.
[14]Gyöngy, I., Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise I, Potential Anal., vol. 9 (1998), pp. 125.
[15]Gyöngy, I. and Mart´inez, T., On numerical solution of stochastic partial differential equations of elliptic type, Stochatics, vol. 78 (2006), pp. 213231.
[16]Jentzen, A. and Kloeden, P.E., The numerical approximation of stochastic partical differential equations, Milan J. Math., vol. 77 (2009), pp. 205244.
[17]Kim, J.U., On a stochatic plate equation, Appl. Math. Optim., vol. 44 (2001), pp. 3348.
[18]Kloeden, P. and Platen, E., Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.
[19]Kloeden, P., Platen, E., and Hoffmann, N., Extrapolation methods for the weak approximation to Itô diffusions, SIAM J. Numer. Anal., vol. 32 (1995), pp. 15191534.
[20]Kossioris, G.T. and Zouraris, G.E., Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise: II. 2D and 3D Case, arXiv:0906.1828v1 [math.NA]
[21]Kovács, M., Larsson, S. and Lindgren, F., Strong convergence of the finite element method with truncated noise for semilinear parabolic stochastic equations, Numer. Algor., vol. 53 (2010), pp. 309320.
[22]Millet, A. and Morine, Pierre-Luc, On implicit and explicit discretization schemes for parabolic SPDEs in any dimension, Stoch. Proc. Appl., vol. 115 (2005), pp. 10731106.
[23]Martin, A., Prigarin, S.M. and Winkler, G., Exact and fast numerical algorithms for the stochastic wave equation, Int. J. Comput. Math., vol. 80 (2003), pp. 15351541.
[24]Prato, G.D. and Zabczyk, J., Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1992.
[25]Prévôt, C. and Röckner, M., A Concise Course on Stochastic Differential Equations, Lecture Notes in Mathmatics, Springer, Berlin, 2007.
[26]Printems, J., On the discretization in time of parabolic stochastic partial differential equtions, Math. Model. Numer. Anal., vol. 35 (2001), pp. 10551078.
[27]Quer-Sardanyons, L. and Sanz-Solé, M., Space semi-discretizations for a stochastic wave equation, Potential Anal., vol. 24 (2006), pp. 303332.
[28]Rozovskii, B.L., Stochastic evolution systems. Linear Theory and Applications to Nonlinear Filtering, Kluwer Academic Publishers Group, Dordrecht, 1990.
[29]Talay, D. and Tubaro, L., Extension of the global error for numerical schemes solving stochastic differential equations, Stoch. Anal. Appl., vol. 8 (1990), pp. 483509.
[30]Walsh, J.B., An Introduction to Stochastic Partial Differential Equations, Lecture Notes in Mathematics, 1180, Springer-verlag, Berlin, (1986), pp. 265439.
[31]Yan, Y., Galerkin finite element methods for stochastic parabolic partial differential equations, SIAM J. Numer. Anal., vol. 43 (2005), pp. 13631384.
[32]Yoo, H., Semi-discretization of stochastic partial differential equations on ℝ1 by a finitedifference method, Math. Comput., vol. 69 (2000), pp. 653666.
[33]Zhang, T., Large deviations for stochastic nonlinear beam equations, J. Funct. Anal., vol. 248 (2007), pp. 175201.

Keywords

MSC classification

Related content

Powered by UNSILO

Difference Approximation of Stochastic Elastic Equation Driven by Infinite Dimensional Noise

  • Yinghan Zhang (a1), Xiaoyuan Yang (a1) and Ruisheng Qi (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.