The Numerical Approximation of Stochastic Partial Differential Equations A. Jentzen and P.E. Kloeden Abstract. The numerical solution of stochastic partial differential equa- tions (SPDEs) is at a stage of development roughly similar to that of stochastic ordinary differential equations (SODEs) in the 1970s, when stochastic Taylor schemes based on an iterated application of the Itˆ o formula were introduced and used to derive higher order numerical schemes. An Itˆo formula in the generality needed for Taylor expansions of the solution of a SPDE is however not available. Nevertheless, it was shown recently how stochastic Taylor expansions for the solution of a SPDE can be derived from the mild form representation of the SPDE, which avoid the need of an Itˆ o formula. A brief review of the literature is given here and the new stochastic Taylor expansions are discussed along with numerical schemes that are based on them. Both strong and pathwise convergence are considered. Mathematics Subject Classification (2000). Primary 60H35; Secondary 60H15. Keywords. Stochastic partial differential equations, stochastic ordinary differential equations, stochastic Taylor expansions, higher order numer- ical schemes, strong convergence, pathwise convergence. This work has been supported by the DFG project “Pathwise numerics and dynamics of stochastic evolution equations” and the spanish Ministerio de Educaci´on y Ciencia project MTM2005-01412. Milan J. Math. Vol. 77 (2009) 205–244 DOI 10.1007/s00032-009-0100-0 Published online September 11, 2009 © 2009 Birkhäuser Verlag Basel/Switzerland Milan Journal of Mathematics