TY - BOOK

T1 - Numerical approximations of nonlinear stochastic partial differential equations appearing in fluid dynamics

AU - Randrianasolo, Tsiry Avisoa

N1 - no embargo

PY - 2018

Y1 - 2018

N2 - In this dissertation, we analyze various discretization of recent mathematical models for turbulent ow modeling. These models share the same complexity. Indeed, they are partial dierential, stochastic, and nonlinear equations. By nonlinear, we mean the equations involve terms which are non-globally Lipschitz or/and non-monotone. And stochastic means, we add noise into the model to capture some disturbances which are inherent in nature. These make the model even more realistic. The results in this work would serve scientist to choose the appropriate numerical methods for their simulations. In the rst part of this dissertation, we consider a stochastic evolution equation in its abstract form. The noise added is a multiplicative noise dened in an innite Hilbert space. The nonlinear term is non-monotone. Models which fall into this abstract equation are the GOY and Sabra shell models and also nonlinear heat equation, of course in presence of noise. The numerical approximation is based on a semi and fully implicit Euler{Maruyama schemes for the time discretization and a spectral Galerkin method for the space discretization. Our result shows a convergence with rate in probability. In the second part, we address the very well-known Navier{Stokes equations with an additive noise. A projection method based on the penalized form of the equation is used. We consider only time-discretization since dierent technicalities appearing after a space-discretization may obscure the main diculty of the projection method. This method breaks the saddle point character of the Navier{Stokes system which is now a sequence of equations much easier to solve. We show the convergence with rate in probability of the scheme for both variables: velocity and pressure. In addition, we also prove the strong convergence of the scheme.

AB - In this dissertation, we analyze various discretization of recent mathematical models for turbulent ow modeling. These models share the same complexity. Indeed, they are partial dierential, stochastic, and nonlinear equations. By nonlinear, we mean the equations involve terms which are non-globally Lipschitz or/and non-monotone. And stochastic means, we add noise into the model to capture some disturbances which are inherent in nature. These make the model even more realistic. The results in this work would serve scientist to choose the appropriate numerical methods for their simulations. In the rst part of this dissertation, we consider a stochastic evolution equation in its abstract form. The noise added is a multiplicative noise dened in an innite Hilbert space. The nonlinear term is non-monotone. Models which fall into this abstract equation are the GOY and Sabra shell models and also nonlinear heat equation, of course in presence of noise. The numerical approximation is based on a semi and fully implicit Euler{Maruyama schemes for the time discretization and a spectral Galerkin method for the space discretization. Our result shows a convergence with rate in probability. In the second part, we address the very well-known Navier{Stokes equations with an additive noise. A projection method based on the penalized form of the equation is used. We consider only time-discretization since dierent technicalities appearing after a space-discretization may obscure the main diculty of the projection method. This method breaks the saddle point character of the Navier{Stokes system which is now a sequence of equations much easier to solve. We show the convergence with rate in probability of the scheme for both variables: velocity and pressure. In addition, we also prove the strong convergence of the scheme.

KW - Fluiddynamik

KW - Stochastische Partielle Differentialgleichung

KW - Numerische Analysis

KW - Navier-Stokes Gleichungen

KW - Approximation

KW - Fehlerabschätzung

KW - Turbulenz

KW - Navier-Strokes

KW - equations

KW - stochastic partial differential equations

KW - fluid dynamics

KW - shell models

KW - turbulence

KW - numerical analysis

KW - error estimates

KW - strong convergence

KW - convergence in probability

KW - approximations

M3 - Doctoral Thesis

ER -