188 research outputs found
Numerical Approximation of Stationary Distribution for SPDEs
In this paper, we show that the exponential integrator scheme both in spatial
discretization and time discretization for a class of stochastic partial
differential equations has a unique stationary distribution whenever the
stepsize is sufficiently small, and reveal that the weak limit of the law for
the exponential integrator scheme is in fact the counterpart for the stochastic
partial differential equation considered.Comment: P2
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
Numerical Solution of Stochastic Partial Differential Equations with Correlated Noise
In this paper we investigate the numerical solution of stochastic partial
differential equations (SPDEs) for a wider class of stochastic equations. We
focus on non-diagonal colored noise instead of the usual space-time white
noise. By applying a spectral Galerkin method for spatial discretization and a
numerical scheme in time introduced by Jentzen Kloeden, we obtain the rate
of path-wise convergence in the uniform topology. The main assumptions are
either uniform bounds on the spectral Galerkin approximation or uniform bounds
on the numerical data. Numerical examples illustrate the theoretically
predicted convergence rate
Galerkin approximations for the stochastic Burgers equation
Existence and uniqueness for semilinear stochastic evolution equations with
additive noise by means of finite dimensional Galerkin approximations is
established and the convergence rate of the Galerkin approximations to the
solution of the stochastic evolution equation is estimated. These abstract
results are applied to several examples of stochastic partial differential
equations (SPDEs) of evolutionary type including a stochastic heat equation, a
stochastic reaction diffusion equation and a stochastic Burgers equation. The
estimated convergence rates are illustrated by numerical simulations. The main
novelty in this article is to estimate the difference of the finite dimensional
Galerkin approximations and of the solution of the infinite dimensional SPDE
uniformly in space, i.e., in the L^{\infty}-topology, instead of the usual
Hilbert space estimates in the L^2-topology, that were shown before.Comment: 22 page
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