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An Exponential Integrator Scheme for Time Discretization of Nonlinear Stochastic Wave Equation

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Abstract

This work is devoted to convergence analysis of an exponential integrator scheme for semi-discretization in time of nonlinear stochastic wave equation. A unified framework is first set forth, which covers important cases of additive and multiplicative noises. Within this framework, the proposed scheme is shown to converge uniformly in the strong \(L^p\)-sense with precise convergence rates given. The abstract results are then applied to several concrete examples. Further, weak convergence rates of the scheme are examined for the case of additive noise. To analyze the weak error for the nonlinear case, techniques based on the Malliavin calculus were usually exploited in the literature. Under certain appropriate assumptions on the nonlinearity, this paper provides a weak error analysis, which does not rely on the Malliavin calculus. The rates of weak convergence can, as expected, be improved in comparison with the strong rates. Both strong and weak convergence results obtained here show that the proposed method achieves higher convergence rates than the implicit Euler and Crank–Nicolson time discretizations. Numerical results are finally reported to confirm our theoretical findings.

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Acknowledgments

The author thanks the anonymous referee whose insightful comments and valuable suggestions are crucial to the improvements of the manuscript. The author would like to thank Professor Arnulf Jentzen for his financial support and helpful discussions during the author’s short visit to ETH Zürich in 2013. Thanks also go to Dr. Fengze Jiang for his careful reading the early version of this manuscript. This work was partially supported by National Natural Science Foundations of China under Grant Numbers 11301550, 11171352, China Postdoctoral Science Foundation under Grant Numbers 2013M531798, 2014T70779 and Research Foundation of Central South University.

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  1. School of Mathematics and Statistics, School of Geosciences and Info-Physics, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Xiaojie Wang

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  1. Xiaojie Wang
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Wang, X. An Exponential Integrator Scheme for Time Discretization of Nonlinear Stochastic Wave Equation. J Sci Comput 64, 234–263 (2015). https://doi.org/10.1007/s10915-014-9931-0

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