REDUCED ORDER METHODS FOR UNCERTAINTY QUANTIFICATION IN COMPUTATIONAL FLUID DYNAMICS.

This master thesis deals with the Stokes and the Navier-Stokes problem in a reduced order framework and its extension to uncertainty quantification. After their success in computational fluid dynamics, one of the problems that emerged is that often the numerical simulations take too much computational time. Usually these problems are relevant when the equations depend on some physical/geometrical parameters and we are interested in the solution for several such parameters, as in the many-query problems or real-time simulation problems. In these cases the finite element method or finite volume method, called full order method, are too slow and we need something faster. One of the solutions for these problems is to use the reduced order method in which the idea is to reconstruct fastly the solution for a certain parameter by a linear combination of precomputed solutions obtained with other parameters, knocking down the computation cost, introducing nevertheless an additional error to the approximation. The uncertainty quantification is introduced in the framework of the Stokes and Navier-Stokes problem because we have said that they can depend on several geometrical or physical parameters that in general could be aftected by some uncertainty and so randomness. In our work we have written a stochastic formulation of the Stokes and Navier-Stokes equations, treating the randomness as parameters. When we are working in a probabilistic framework we can use the information related to the different distributions associated to the parameters introducing some weights in the two algorithms mentioned before, to catch the most likely parameters in several way, needed for calculating the solutions used in the linear combination. We note that when we introduce uncertainties we can use some methods for chosing the parameters, such as the tensor product rule, the Monte-Carlo method, both usually expensive from a computational cost point of view, and the Smolyak rule, cheaper but less precise that the previous ones. We have done some numerical experiments using RBniCS library, developed at SISSA mathLab. We have studied some problems associated to several different probabilistic distributions, observing the strong points and the weaknesses associated to the methods that we have treated in the theoretical part.