Abstract
This paper aims to investigate the weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. We are interested in SPDEs where the nonlinear part is stronger than the linear part, also called stochastic reaction dominated transport equations. For such SPDEs, many standard numerical schemes lose their stability properties. Exponential Rosenbrock and Rosenbrock-type methods were proved to be efficient for such SPDEs, but only their strong convergence were recently analyzed. Here, we investigate the weak convergence of the Rosenbrock semi-implicit method. We obtain a weak convergence rate which is twice the rate of the strong convergence. Our error analysis does not rely on Malliavin calculus, but rather only uses the Kolmogorov equation and the smoothing properties of the resolvent operator resulting from the Rosenbrock semi-implicit approximation.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: 317210226
Funding statement: The work of J. D. Mukam was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 317210226 – SFB 1283.
Acknowledgements
The authors would like to thank the anonymous referees for their insightful comments that helped to improve this work.
References
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