A High Order Finite Di erence Method for Random Parabolic Partial Di erential Equations


Department of Mathematics, Faculty of Mathematics, Vali{e{Asr University of Rafsanjan, Rafsanjan. Iran.


In this paper, for the numerical approximation of random partial differential equations (RPDEs) of parabolic type, an explicit higher order finite difference scheme is constructed. In continuation the main properties of deterministic difference schemes, i.e. consistency, stability and convergency are developed for the random cases. It is shown that the proposed random difference scheme has these properties. Finally a numerical example is solved to illustrate the scheme of analysis.


1. E. Allen, S.L. Novose and Z. Zhang, Finite element and difference approximation of some
linear stochastic partial differential equations, Stochastic Rep. 64 (1998), 117-142.
2. A. Barth, A _nite element method for martingale driven stochastic partial differential
equations, Comm. Stoch. Anal. (3) (2010), 355{375.
3. A. Barth, Stochastic partial differential equations: approximations and applications, Ph.
D. thesis, University of Oslo, CMA, September 2009.
4. J. C. Cortes et al., Computing mean square approximations of random differential models, Math. Comput. Simulat. 76 (2007), 44-48.
5. J. C. Cortes et al., Mean square convergent numerical methods, Trans. Comput. Sci.VII 5890 (2010), 1-21.
6. C. Roth, Di_erence methods for stochastic partial di_erential equations, Z. Zngew. Math. Mech. 82 (2002), 821-830.
7. T. T. Soong, Random differential equations in science and engineering, Academic Press, New York, 1973.
8. J. Thomas, Numerical partial differential equations: finite difference methods, Texts in Applied Mathematics, Springer, 1998.