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Volume 14, Issue 1
Mean-Square Approximation of Navier-Stokes Equations with Additive Noise in Vorticity-Velocity Formulation

G.N. Milstein & M.V. Tretyakov

DOI: 10.4208/nmtma.OA-2020-0034

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 1-30.

Published online: 2020-10

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  • Abstract

We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions and additive noise in the vorticity-velocity formulation. The approximation is based on freezing the velocity on time subintervals resulting in a linear stochastic parabolic equation for vorticity. At each time step, the velocity is expressed via vorticity using a formula corresponding to the Biot-Savart-type law. We prove the first mean-square convergence order of the vorticity approximation.

  • Keywords

Navier-Stokes equations, vorticity, numerical method, stochastic partial differential equations, mean-square convergence.

  • AMS Subject Headings

65C30, 60H15, 60H35, 35Q30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-14-1, author = {Milstein , G.N. and Tretyakov , M.V.}, title = {Mean-Square Approximation of Navier-Stokes Equations with Additive Noise in Vorticity-Velocity Formulation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {14}, number = {1}, pages = {1--30}, abstract = {

We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions and additive noise in the vorticity-velocity formulation. The approximation is based on freezing the velocity on time subintervals resulting in a linear stochastic parabolic equation for vorticity. At each time step, the velocity is expressed via vorticity using a formula corresponding to the Biot-Savart-type law. We prove the first mean-square convergence order of the vorticity approximation.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2020-0034}, url = {http://global-sci.org/intro/article_detail/nmtma/18325.html} }
TY - JOUR T1 - Mean-Square Approximation of Navier-Stokes Equations with Additive Noise in Vorticity-Velocity Formulation AU - Milstein , G.N. AU - Tretyakov , M.V. JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 1 EP - 30 PY - 2020 DA - 2020/10 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2020-0034 UR - https://global-sci.org/intro/article_detail/nmtma/18325.html KW - Navier-Stokes equations, vorticity, numerical method, stochastic partial differential equations, mean-square convergence. AB -

We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions and additive noise in the vorticity-velocity formulation. The approximation is based on freezing the velocity on time subintervals resulting in a linear stochastic parabolic equation for vorticity. At each time step, the velocity is expressed via vorticity using a formula corresponding to the Biot-Savart-type law. We prove the first mean-square convergence order of the vorticity approximation.

Milstein , G.N. and Tretyakov , M.V.. (2020). Mean-Square Approximation of Navier-Stokes Equations with Additive Noise in Vorticity-Velocity Formulation. Numerical Mathematics: Theory, Methods and Applications. 14 (1). 1-30. doi:10.4208/nmtma.OA-2020-0034
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