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Volume 37, Issue 3
Numerical Solutions of Nonautonomous Stochastic Delay Differential Equations by Discontinuous Galerkin Methods

Xinjie Dai & Aiguo Xiao

DOI: 10.4208/jcm.1806-m2017-0296

J. Comp. Math., 37 (2019), pp. 419-436.

Published online: 2018-09

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  • Abstract

This paper considers a class of discontinuous Galerkin method, which is constructed by Wong-Zakai approximation with the orthonormal Fourier basis, for numerically solving nonautonomous Stratonovich stochastic delay differential equations. We prove that the discontinuous Galerkin scheme is strongly convergent, globally stable and analogously asymptotically stable in mean square sense. In addition, this method can be easily extended to solve nonautonomous Stratonovich stochastic pantograph differential equations. Numerical tests indicate that the method has first-order and half-order strong mean square convergence, when the diffusion term is without delay and with delay, respectively.

  • Keywords

Discontinuous Galerkin method, Wong-Zakai approximation, Nonautonomous Stratonovich stochastic delay differential equation.

  • AMS Subject Headings

34K50, 60H35, 65L20, 65L60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

xinjie@smail.xtu.edu.cn (Xinjie Dai)

xag@xtu.edu.cn (Aiguo Xiao)

  • BibTex
  • RIS
  • TXT
@Article{JCM-37-419, author = {Dai , Xinjie and Xiao , Aiguo}, title = {Numerical Solutions of Nonautonomous Stochastic Delay Differential Equations by Discontinuous Galerkin Methods}, journal = {Journal of Computational Mathematics}, year = {2018}, volume = {37}, number = {3}, pages = {419--436}, abstract = {

This paper considers a class of discontinuous Galerkin method, which is constructed by Wong-Zakai approximation with the orthonormal Fourier basis, for numerically solving nonautonomous Stratonovich stochastic delay differential equations. We prove that the discontinuous Galerkin scheme is strongly convergent, globally stable and analogously asymptotically stable in mean square sense. In addition, this method can be easily extended to solve nonautonomous Stratonovich stochastic pantograph differential equations. Numerical tests indicate that the method has first-order and half-order strong mean square convergence, when the diffusion term is without delay and with delay, respectively.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1806-m2017-0296}, url = {http://global-sci.org/intro/article_detail/jcm/12731.html} }
TY - JOUR T1 - Numerical Solutions of Nonautonomous Stochastic Delay Differential Equations by Discontinuous Galerkin Methods AU - Dai , Xinjie AU - Xiao , Aiguo JO - Journal of Computational Mathematics VL - 3 SP - 419 EP - 436 PY - 2018 DA - 2018/09 SN - 37 DO - http://doi.org/10.4208/jcm.1806-m2017-0296 UR - https://global-sci.org/intro/article_detail/jcm/12731.html KW - Discontinuous Galerkin method, Wong-Zakai approximation, Nonautonomous Stratonovich stochastic delay differential equation. AB -

This paper considers a class of discontinuous Galerkin method, which is constructed by Wong-Zakai approximation with the orthonormal Fourier basis, for numerically solving nonautonomous Stratonovich stochastic delay differential equations. We prove that the discontinuous Galerkin scheme is strongly convergent, globally stable and analogously asymptotically stable in mean square sense. In addition, this method can be easily extended to solve nonautonomous Stratonovich stochastic pantograph differential equations. Numerical tests indicate that the method has first-order and half-order strong mean square convergence, when the diffusion term is without delay and with delay, respectively.

Xinjie Dai & Aiguo Xiao. (2019). Numerical Solutions of Nonautonomous Stochastic Delay Differential Equations by Discontinuous Galerkin Methods. Journal of Computational Mathematics. 37 (3). 419-436. doi:10.4208/jcm.1806-m2017-0296
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