Strong convergence of full-discrete nonlinearity-truncated accelerated exponential euler-type approximations for stochastic Kuramoto–Sivashinsky equations

Abstract

This article introduces and analyzes a new explicit, easily implementable, and full discrete accelerated exponential Euler-type approximation scheme for additive space-time white noise driven stochastic partial differential equations (SPDEs) with possibly non-globally monotone nonlinearities such as stochastic Kuramoto-Sivashinsky equations. The main result of this article proves that the proposed approximation scheme converges strongly and numerically weakly to the solution process of such an SPDE. Key ingredients in the proof of our convergence result are a suitable generalized coercivity-type condition, the specific design of the accelerated exponential Euler-type approximation scheme, and an application of Fernique's theorem.

Existence, uniqueness, and numerical approximations for stochastic Burgers equations

“…To conclude, let us mention the fact that our main all-in-one results (in particular, Corollary 3.3 below) can also be applied to the Kuramoto-Sivashinsky equations considered in Hutzenthaler et al [2016], recovering the strong convergence result for the numerical scheme obtained there and also recovering the existence and uniqueness of the mild solution obtained in, e.g., Duan and Ervin [2001].…”

“…Combining this, the fact that lim sup n→∞ P n | H̺ L(H̺) = 1 < ∞, (3.16), the assumption that lim sup n→∞ h n = 0, (3.25), (3.10), and (3.22) allows us to apply Lemma 2.3 (with (V, Moreover, note that Lemma 2.3 in Hutzenthaler et al [2016] and the assumption that O n : [0, T ] × Ω → H ̺ , n ∈ N, are stochastic processes with continuous sample paths ensure that X n : [0, T ] × Ω → H ̺ , n ∈ N, are stochastic processes with rightcontinuous sample paths. This, (3.28), the fact that ∀ t ∈ [0, T ] : X t | Ω\Ω 1 = O t | Ω\Ω 1 , and the fact that Ω 1 ∈ F prove that X : [0, T ] × Ω → H ̺ is a stochastic process.…”

“…Proof of Lemma 2.3. First, observe that Proposition 3.3 in Hutzenthaler et al [2016] shows that for all t ∈ J it holds that lim sup…”

“…Next note that the assumption that X n , O n : [0, T ] × Ω → H ̺ , n ∈ N, are stochastic processes and (3.4) prove that for all n ∈ N it holds that X n : [0, T ] × Ω → H ̺ is also a stochastic process. The assumption that O n : [0, T ] × Ω → H ̺ , n ∈ N, are continuous, and, e.g., Lemma 2.2 in Hutzenthaler et al [2016] therefore ensure that X n : [0, T ] × Ω → H ̺ , n ∈ N, are stochastic processes with right-continuous sample paths. 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).…”

Optimal Error Estimates of Galerkin Finite Element Methods for Stochastic Allen–Cahn Equation with Additive Noise

“…Over the past decades, there have been plenty of research articles analyzing numerical discretizations of parabolic stochastic partial differential equations (SPDEs), see, e.g., monographs [21,25] and references therein. In contrast to an overwhelming majority of literature focusing on numerical analysis of SPDEs with globally Lipschitz nonlinearity, only a limited number of papers investigated numerical SPDEs in the non-globally Lipschitz regime [1][2][3][4][5]10,11,13,15,16,19,24] and it is still far from well-understood. As a typical example of parabolic SPDEs with non-globally Lipschitz nonlinearity, stochastic Allen-Cahn equations, perturbed by additive or multiplicative noises, have received increasing attention in the last few years.…”