Strong convergence for explicit space–time discrete numerical approximation methods for stochastic Burgers equations

Abstract

In this paper we propose and analyze explicit space-time discrete numerical approximations for additive space-time white noise driven stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities such as the stochastic Burgers equation with space-time white noise. The main result of this paper proves that the proposed explicit space-time discrete approximation method converges strongly to the solution process of the stochastic Burgers equation with space-time white noise. To the best of our knowledge, the main result of this work is the first result in the literature which establishes strong convergence for a space-time discrete approximation method in the case of the stochastic Burgers equations with space-time white noise.

Existence, uniqueness, and numerical approximations for stochastic Burgers equations

“…Next observe that the fact that H ⊆ H −1 =H H −1 and the fact that for all n ∈ N it holds that P n ∈ L(H) imply that there existP n ∈ L(H −1 , H), n ∈ N, such that for all v ∈ H, n ∈ N it holds that P n (v) = P n (v). Items (i),(ii) and (iii) in Theorem 3.5 in Jentzen et al [2019] (with P n =P n , n ∈ N, in the notation of Theorem 3.5 in Jentzen et al [2019]) therefore establish Items (ii),(iii), and (iv). The proof of Corollary 3.3 is thus completed.…”

“…Next note that the assumption that sup b∈H |λ b | < ∞ assures that A ∈ L(H). Corollary 2.6 in Jentzen et al [2019] therefore ensures that ηO ∈ C([0, T ], H) and that…”

“…Then note that Lemma 6.3 in Jentzen et al [2019] shows that for all v, w ∈ H1 /2 it holds that F (v + w) ∈ H and that v, F (v + w) H ≤ max 2|c 1 | 2 c 0 , 4 v 2 H sup x∈(0,1) |w(x)| 2 + 3 4 (−A) 1 /2 v 2 H + max 2|c 1 | 2 c 0 , 4 1 +sup x∈(0,1) |w(x)| max{ 2|c 1 | 2 /c 0 ,4} ≤ φ(w) v 2 H + 3 4 (−A) 1 /2 v 2 H + Φ(w).…”

“…To explain our result better let us consider H to be the real Hilbert space given by H = L 2 ((0, 1); R), A : D(A) ⊆ H → H to be the Laplace operator with Dirichlet boundary conditions on H, and (H r , ·, · Hr , · Hr ), r ∈ R, to be a family of interpolation spaces associated to −A. The main result of this paper, Theorem 3.2 below, is applicable to a subclass of stochastic evolution equations considered in Theorem 3.5 in Jentzen et al [2019]. This subclass has to satisfy an additional regularity condition on the nonlinearity (see Setting 3.1, in particular, inequality (3.1) below), which is crucial in the proof of pathwise a priori estimates for the approximation process (see Lemma 2.2 below).…”

Convergence Rate for Galerkin Approximation of the Stochastic Allen—Cahn Equations on 2D Torus

“…[2,7,10,17,19,22,24,26,39]). For SPDEs driven by space-time white noise, we refer to [5,8,11,12,20,21,23] and the references therein for the convergence of spatial approximations and refer to [3,5,8,11,12,21,23] and the references therein for the convergence of temporal approximations. Furthermore, in the references [3,5,8,12,20] the convergence rates of spatial and temporal approximations were also obtained.…”