We consider the numerical approximation of stochastic differential equations interpreted both in the It^o and Stratonovich sense and develop three stochastic time-integration techniques based on the deterministic exponential time differencing schemes. Two of the numerical schemes are suited for the simulations of It^o stochastic ordinary differential equations (SODEs) and they are referred to as the stochastic exponential time differencing schemes, SETD0 and SETD1. The third numerical scheme is a new numerical method we propose for the simulations of Stratonovich SODEs. We call this scheme, the Exponential Stratonovich Integrator (ESI). We investigate numerically the convergence of these three numerical methods, in addition to three standard approximation schemes and also compare the accuracy and efficiency of these schemes. The effect of small noise is also studied. We study the theoretical convergence of the stochastic exponential time differencing scheme (SETD0) for parabolic stochastic partial differential equations (SPDEs) with infinite-dimensional additive noise and one-dimensional multiplicative noise. We obtain a strong error temporal estimate of O(¢tµ + ²¢tµ + ²2¢t1=2) for SPDEs forced with a one-dimensional multiplicative noise and also obtain a strong error temporal estimate of O(¢tµ + ²2¢t) for SPDEs forced with an infinite-dimensional additive noise. We examine convergence for second-order and fourth-order SPDEs. We consider the effects of spatially correlated and uncorrelated noise on bifurcations for SPDEs. In particular, we study a fourth-order SPDE, the Swift-Hohenberg equation, and allow the control parameter to fluctuate. Numerical simulations show a shift in the pinning region with multiplicative noise.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:575244 |
| Date | January 2011 |
| Creators | Adamu, Iyabo Ann |
| Contributors | Lord, Gabriel L. |
| Publisher | Heriot-Watt University |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/10399/2460 |
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