Volume 13, Issue 1
Auxiliary Equations Approach for the Stochastic Unsteady Navier-Stokes Equations with Additive Random Noise

Wenju Zhao & Max Gunzburger

Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 1-26.

Published online: 2019-12

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  • Abstract

This paper presents a Martingale regularization method for the stochastic Navier-Stokes equations with additive noise. The original system is split into two equivalent parts, the linear stochastic Stokes equations with Martingale solution and the stochastic modified Navier-Stokes equations with relatively-higher regularities. Meanwhile, a fractional Laplace operator is introduced to regularize the noise term. The stability and convergence of numerical scheme for the pathwise modified Navier-Stokes equations are proved. The comparisons of non-regularized and regularized noises for the Navier-Stokes system are numerically presented to further demonstrate the efficiency of our numerical scheme.

  • Keywords

Stochastic Navier--Stokes equations, Martingale regularization method, Galerkin finite element method, white noise.

  • AMS Subject Headings

35R60, 65Mxx, 76Dxx

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

wz13@my.fsu.edu (Wenju Zhao)

mgunzburger@fsu.edu (Max Gunzburger)

  • BibTex
  • RIS
  • TXT
@Article{NMTMA-13-1, author = {Zhao , Wenju and Gunzburger , Max }, title = {Auxiliary Equations Approach for the Stochastic Unsteady Navier-Stokes Equations with Additive Random Noise}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {13}, number = {1}, pages = {1--26}, abstract = {

This paper presents a Martingale regularization method for the stochastic Navier-Stokes equations with additive noise. The original system is split into two equivalent parts, the linear stochastic Stokes equations with Martingale solution and the stochastic modified Navier-Stokes equations with relatively-higher regularities. Meanwhile, a fractional Laplace operator is introduced to regularize the noise term. The stability and convergence of numerical scheme for the pathwise modified Navier-Stokes equations are proved. The comparisons of non-regularized and regularized noises for the Navier-Stokes system are numerically presented to further demonstrate the efficiency of our numerical scheme.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2019-0055}, url = {http://global-sci.org/intro/article_detail/nmtma/13428.html} }
TY - JOUR T1 - Auxiliary Equations Approach for the Stochastic Unsteady Navier-Stokes Equations with Additive Random Noise AU - Zhao , Wenju AU - Gunzburger , Max JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 1 EP - 26 PY - 2019 DA - 2019/12 SN - 13 DO - http://dor.org/10.4208/nmtma.OA-2019-0055 UR - https://global-sci.org/intro/article_detail/nmtma/13428.html KW - Stochastic Navier--Stokes equations, Martingale regularization method, Galerkin finite element method, white noise. AB -

This paper presents a Martingale regularization method for the stochastic Navier-Stokes equations with additive noise. The original system is split into two equivalent parts, the linear stochastic Stokes equations with Martingale solution and the stochastic modified Navier-Stokes equations with relatively-higher regularities. Meanwhile, a fractional Laplace operator is introduced to regularize the noise term. The stability and convergence of numerical scheme for the pathwise modified Navier-Stokes equations are proved. The comparisons of non-regularized and regularized noises for the Navier-Stokes system are numerically presented to further demonstrate the efficiency of our numerical scheme.

Wenju Zhao & Max Gunzburger. (2019). Auxiliary Equations Approach for the Stochastic Unsteady Navier-Stokes Equations with Additive Random Noise. Numerical Mathematics: Theory, Methods and Applications. 13 (1). 1-26. doi:10.4208/nmtma.OA-2019-0055
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