Numerical approximation of SDEs and stochastic Swift-Hohenberg equation

Abstract

We consider the numerical approximation of stochastic di®erential equations inter- preted both in the It^o and Stratonovich sense and develop three stochastic time- integration techniques based on the deterministic exponential time di®erencing schemes. Two of the numerical schemes are suited for the simulations of It^o stochastic ordinary di®erential equations (SODEs) and they are referred to as the stochastic exponential time di®erencing 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 ad- dition to three standard approximation schemes and also compare the accuracy and e±ciency of these schemes. The e®ect of small noise is also studied. We study the theoretical convergence of the stochastic exponential time di®erencing scheme (SETD0) for parabolic stochastic partial di®erential equations (SPDEs) with in¯nite-dimensional additive noise and one-dimensional multiplicative noise. We ob- tain 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 in¯nite-dimensional additive noise. We examine convergence for second-order and fourth-order SPDEs. We consider the e®ects of spatially correlated and uncorrelated noise on bifurcations for SPDEs. In particular, we study a fourth-order SPDE, the Swift-Hohenberg equa- tion, and allow the control parameter to °uctuate. Numerical simulations show a shift in the pinning region with multiplicative noise.