117 research outputs found
A modified semi--implict Euler-Maruyama Scheme for finite element discretization of SPDEs with additive noise
We consider the numerical approximation of a general second order
semi--linear parabolic stochastic partial differential equation (SPDE) driven
by additive space-time noise. We introduce a new modified scheme using a linear
functional of the noise with a semi--implicit Euler--Maruyama method in time
and in space we analyse a finite element method (although extension to finite
differences or finite volumes would be possible). We prove convergence in the
root mean square norm for a diffusion reaction equation and diffusion
advection reaction equation. We present numerical results for a linear reaction
diffusion equation in two dimensions as well as a nonlinear example of
two-dimensional stochastic advection diffusion reaction equation. We see from
both the analysis and numerics that the proposed scheme has better convergence
properties than the standard semi--implicit Euler--Maruyama method
A Milstein scheme for SPDEs
This article studies an infinite dimensional analog of Milstein's scheme for
finite dimensional stochastic ordinary differential equations (SODEs). The
Milstein scheme is known to be impressively efficient for SODEs which fulfill a
certain commutativity type condition. This article introduces the infinite
dimensional analog of this commutativity type condition and observes that a
certain class of semilinear stochastic partial differential equation (SPDEs)
with multiplicative trace class noise naturally fulfills the resulting infinite
dimensional commutativity condition. In particular, a suitable infinite
dimensional analog of Milstein's algorithm can be simulated efficiently for
such SPDEs and requires less computational operations and random variables than
previously considered algorithms for simulating such SPDEs. The analysis is
supported by numerical results for a stochastic heat equation and stochastic
reaction diffusion equations showing signifficant computational savings.Comment: The article is slightly revised and shortened. In particular, some
numerical simulations are remove
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