A Newton method for the resolution of steady stochastic Navier-Stokes equations |
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Authors: | Olivier Le Maître |
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Affiliation: | a LIMSI-CNRS, Mechanical Engineering, BP133, F-91403 Orsay Cedex, France b CEA-DM2S, Centre de Saclay, 91 191 Gif sur Yvette, France |
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Abstract: | We present a Newton method to compute the stochastic solution of the steady incompressible Navier-Stokes equations with random data (boundary conditions, forcing term, fluid properties). The method assumes a spectral discretization at the stochastic level involving a orthogonal basis of random functionals (such as Polynomial Chaos or stochastic multi-wavelets bases). The Newton method uses the unsteady equations to derive a linear equation for the stochastic Newton increments. This linear equation is subsequently solved following a matrix-free strategy, where the iterations consist in performing integrations of the linearized unsteady Navier-Stokes equations, with an appropriate time scheme to allow for a decoupled integration of the stochastic modes. Various examples are provided to demonstrate the efficiency of the method in determining stochastic steady solution, even for regimes where it is likely unstable. |
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