Numerical approximation of multiplicative SPDEs

Abstract

We prove convergence of the stochastic exponential time differencing scheme for parabolic stochastic partial differential equations (SPDEs) with one-dimensional multiplicative noise. We examine convergence for fourth-order SPDEs and consider as an example the Swift–Hohenberg equation. After examining convergence, we present preliminary evidence of a shift in the deterministic pinning region [J. Burke and E. Knobloch, Localized states in the generalized Swift–Hohenberg equation, Phys. Rev. E. 73 (2006), pp. 056211-1–15; J. Burke and E. Knobloch, Snakes and ladders: Localized states in the Swift–Hohenberg equation, Phys. Lett. A 360 (2007), pp. 681–688; Y.-P. Ma, J. Burke, and E. Knobloch, Snaking of radial solutions of the multi-dimensional Swift–Hohenberg equation: A numerical study, Physica D 239 (2010), pp. 1867–1883; S. McCalla and B. Sandstede, Snaking of radial solutions of the multi-dimensional Swift–Hohenberg equation: A numerical study, Physica D 239 (2010), pp. 1581–1592] with space–time white noise.

Keywords:

• stochastic partial differential equations
• multiplicative noise
• stochastic exponential time differencing
• Swift–Hohenberg equation
• pinning region

2011 AMS Subject Classifications:

• 35R60
• 60H15
• 60H35
• 65M15
• 65M60