Numerical study of amplitude equations for SPDEs with degenerate forcing

Abstract

In this paper, we give a review on rigorous and numerical results for amplitude equations. We focus on the Swift–Hohenberg equation and the Burgers’ equation in order to determine the quality of the approximation and the impact of degenerate noise on the approximating equation.

Keywords:

• SPDE
• amplitude equation
• degenerate noise
• spectral Galerkin method
• multi-scale analysis

2010 AMS Subject Classification :

• 60H15
• 60H10
• 60H35
• 65M70
• 65C05

Acknowledgements

This work was supported by Deutsche Forschungsgemeinschaft (DGF) ‘Multi-scale analysis of SPDEs’ (BL535-9/1). We also thank the anonymous referee for many valuable comments.