# A Milstein Scheme for SPDEs

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## Abstract

This article studies an infinite-dimensional analog of Milstein’s scheme for finite-dimensional stochastic ordinary differential equations (SODEs). The Milstein scheme is known to be impressively efficient for SODEs which fulfill a certain commutativity type condition. This article introduces the infinite-dimensional analog of this commutativity type condition and observes that a certain class of semilinear stochastic partial differential equation (SPDEs) with multiplicative trace class noise naturally fulfills the resulting infinite-dimensional commutativity condition. In particular, a suitable infinite-dimensional analog of Milstein’s algorithm can be simulated efficiently for such SPDEs and requires less computational operations and random variables than previously considered algorithms for simulating such SPDEs. The analysis is supported by numerical results for a stochastic heat equation, stochastic reaction diffusion equations and a stochastic Burgers equation, showing significant computational savings.

## Keywords

Milstein scheme Stochastic partial differential equation SPDE Stochastic differential equation SDE Numerical approximation Higher-order approximation## Mathematics Subject Classification

60H35 65C30## Notes

### Acknowledgments

This work has been supported by the BiBoS Research Center, by the research project “Numerical solutions of stochastic differential equations with non-globally Lipschitz continuous coefficients” funded by the German Research Foundation, by the Collaborative Research Centre 701 “Spectral Structures and Topological Methods in Mathematics” funded by the German Research Foundation and by the International Graduate School “Stochastics and Real World Models” funded by the German Research Foundation. The support of Issac Newton Institute for Mathematical Sciences in Cambridge is also gratefully acknowledged where part of this was done during the special semester on “Stochastic Partial Differential Equations”. In addition, we are very grateful to Sebastian Becker for his help with the numerical simulations and to Carlo Marinelli for his helpful advice concerning Schatten norms.

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