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

TL;DR

This paper develops and proves the strong convergence of a new explicit accelerated exponential Euler-type scheme for stochastic Kuramoto-Sivashinsky equations driven by space-time white noise, accommodating non-globally monotone nonlinearities.

Contribution

It introduces a novel full-discrete approximation scheme that is easily implementable and proves its strong convergence for complex SPDEs with non-globally monotone nonlinearities.

Findings

The scheme converges strongly to the SPDE solution.

The scheme also converges weakly in a numerical sense.

The proof relies on a generalized coercivity condition and Fernique's theorem.

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.