Numerical approximation of Stratonovich SDEs and SPDEs
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We consider the numerical approximation of stochastic differential and partial differential equations S(P)DEs, by means of time-differencing schemes which are based on exponential integrator techniques. We focus on the study of two numerical schemes, both appropriate for the simulation of Stratonovich- interpreted S(P)DEs. The first, is a basic strong order 1=2 scheme, called Stratonovich Exponential Integrators (SEI). Motivated by SEI and aiming at benefiting both from the higher order of the standard Milstein scheme and the efficiency of the exponential schemes when dealing with stiff problems, we develop a new Milstein type scheme called Milstein Stratonovich Exponential Integrators (MSEI). We prove strong convergence of the SEI scheme for high-dimensional semilinear Stratonovich SDEs with multiplicative noise and we use SEI as well as the MSEI scheme to approximate solutions of the stochastic Landau-Lifschitz- Gilbert (LLG) equation in three dimensions. We examine the L2(Ω ) approximation error of the SEI and MSEI schemes numerically and we prove analytically that MSEI achieves a higher order of convergence than SEI. We generalise SEI so that it is suited not only for Stratonovich SDEs, but also for It^o and for SDEs interpreted by the 'in-between' calculi. Moreover, we provide a general expression for the predictor contained in SEI and we study the theoretical convergence for the generalised version of the scheme. We show that the order of the scheme used in order to obtain the predictor as well as the stochastic integral interpretation do not affect the overall order of the scheme. We extend the convergence results for SEI to a space-time context by considering a second order semilinear Stratonovich SPDE with multiplicative noise. We discretise in space with the nite element method and we use SEI for discretising in time. We consider the case where we have trace class noise and we examine analytically the strong order of convergence for SEI. We implement SEI as a time discretisation scheme and present the results when simulating SPDEs with stochastic travelling wave solutions. Then, we use an alternative method, called 'freezing' method, for approximating wave solutions and estimating the speed of the waves for the stochastic Nagumo and FitzHugh-Nagumo models. The wave position and hence the speed is found by minimising the L2 distance between a reference function and the travelling wave. While the results obtained from the two different approaches agree, we observe that the behaviour of the wave solution is captured in a smaller computational domain, when we use the freezing method, making it more efficient for long time simulations.