Communications in Mathematical Sciences

Volume 16 (2018)

Number 6

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

Pages: 1489 – 1529

DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n6.a2

Authors

Martin Hutzenthaler (Faculty of Mathematics, University of Duisburg, Essen, Germany)

Arnulf Jentzen (Seminar for Applied Mathematics, Department of Mathematics, ETH Zürich, Switzerland)

Diyora Salimova (Seminar for Applied Mathematics, Department of Mathematics, ETH Zürich, Switzerland)

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.

Keywords

stochastic differential equation, strong convergence, numerical approximation, stochastic Kuramoto–Sivashinsky equations, coercivity-type condition, accelerated exponential Euler approximations

2010 Mathematics Subject Classification

60H35, 65C30

Full Text (PDF format)

Received 21 May 2017

Accepted 14 January 2018

Published 7 February 2019