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

Tsiry Avisoa Randrianasolo

Research output: ThesisDoctoral Thesis

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Abstract

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.
Translated title of the contributionNumerische Approximation nichtlinearer stochastischer partieller Differentialgleichungen in der Fluiddynamik
Original languageEnglish
QualificationDr.mont.
Awarding Institution
  • Montanuniversität
Supervisors/Advisors
  • Hausenblas, Erika, Assessor A (internal)
  • Banas, Lubomir, Assessor B (external), External person
Publication statusPublished - 2018

Bibliographical note

embargoed until null

Keywords

  • Navier-Strokes
  • equations
  • stochastic partial differential equations
  • fluid dynamics
  • shell models
  • turbulence
  • numerical analysis
  • error estimates
  • strong convergence
  • convergence in probability
  • approximations

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