Shardlow, Tony Numerical methods for stochastic parabolic PDEs. (English) Zbl 0919.65100 Numer. Funct. Anal. Optimization 20, No. 1-2, 121-145 (1999). This paper presents a proof of the convergence of finite difference approximations of the solution of the nonlinear stochastic partial differential equation initial value problem of the form \[ du(t)= \Biggl[{\partial^2u(t)\over\partial x^2}+ f(u(t))\Biggr] dt+ dB(t),\quad u(0)= U, \] where \(B(t)\) is a Wiener process. It concludes with a brief summary of results obtained in numerical experiments with \(f=0\) and with \(f= .5(u- u^3)\). Reviewer: M.D.Lax (Long Beach) Cited in 50 Documents MSC: 65C99 Probabilistic methods, stochastic differential equations 35K55 Nonlinear parabolic equations 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35R60 PDEs with randomness, stochastic partial differential equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs Keywords:convergence; finite difference; nonlinear stochastic partial differential equation; initial value problem; Wiener process; numerical experiments PDFBibTeX XMLCite \textit{T. Shardlow}, Numer. Funct. Anal. Optim. 20, No. 1--2, 121--145 (1999; Zbl 0919.65100) Full Text: DOI References: [1] Da Prato G., Encyclopedia of Mathematics and its Applications 44 (1992) [2] Da Prato G., London Mathematical Society Lecture Note Series 229 (1996) [3] Davie A. M., Convergence of implicit schemes for numerical solution of parabolic stochastic partial differential equations (1996) [4] DOI: 10.1090/S0025-5718-1992-1122067-1 [5] Pazy A., Applied Mathematical Sciences 44 (1983) [6] Shardlow T., Stoch. Anal. App. 17 (1999) [7] Shardlow T., SIAM J. Num. Anal. 17 (1999) 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.