Full discretization of the stochastic Burgers equation with correlated noise

Abstract

The main purpose of this paper is to investigate the spectral Galerkin method for spatial discretization. We combine it with the method introduced by Kloeden, Jentzen & Winkel in [12] for temporal discretization of stochastic partial differential equations and study pathwise convergence. We consider the case of colored noise, instead of the usual space-time white noise that was used before for the spatial discretization. The rate of convergence in uniform topology is estimated for the stochastic Burgers equation. Numerical examples illustrate the estimated convergence rate.

Convergence of a numerical scheme for SPDEs with correlated noise and global Lipschitz coefficients

“…From together with Lemma in , we conclude there exists a stochastic process O :[0, T ]×Ω→ V , which is the Ornstein–Uhlenbeck process (or stochastic convolution) given by the semigroup generated by the Dirichlet Laplacian and the Wiener process , i.e, . In addition, from Lemma in we derive that O satisfies Assumption , for all , with …”

“…For the regularity, assume that for some ρ > 0, we have Moreover, assume ξ :Ω→ V is a measurable mapping with for every ω ∈Ω. From together with Lemma in , we conclude there exists a stochastic process O :[0, T ]×Ω→ V , which is the Ornstein–Uhlenbeck process (or stochastic convolution) given by the semigroup generated by the Dirichlet Laplacian and the Wiener process , i.e, . In addition, from Lemma in we derive that O satisfies Assumption , for all , with …”

“…2 . From (21) together with Lemma 3 in[12], we conclude there exists a stochastic process O : OE0, T ! V, which is the Ornstein-Uhlenbeck process (or stochastic convolution) given by the semigroup generated by the Dirichlet Laplacian and the Wiener process W.t/ D P…”

Implicit Euler method for numerical solution of nonlinear stochastic partial differential equations with multiplicative trace class noise

“…6 Printems used finite difference method for time discretization based on the spectral properties of the linear operator in the SPDE. 7 Spectral Galerkin method has been extensively studied for SPDEs [8][9][10][11][12][13][14][15] with mostly additive noise. Some papers concern with spectral Galerkin method in the case of colored noise.…”

On the Backward Euler Approximation of the Stochastic Allen-Cahn Equation

“…For non-global Lipschitz nonlinearities the relatively recent method developed in [12] uses a scheme which is based on the mild formulation of the SPDE. This is also employed, for example, in [2]. In that setting pathwise error estimates are derived but strong convergence results would be rather difficult to obtain as the method loses the information about the one-sided Lipschitz condition on f , which can only be exploited in a variational or weak solution approach.…”