No abstract

In this work, we propose viable and efficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of wRB (weighted reduced basis) method for stochastic parametrized problems with stabilized reduced basis method, which is the integration of classical stabilization methods (SUPG, in our case) in the Offline-Online structure of the RB method. Moreover, we introduce a reduction method that selectively enables online stabilization; this leads to a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We present numerical test cases to assess the performance of the proposed methods in steady and unsteady problems related to heat transfer phenomena.Key words. random inputs, reduced basis methods, uncertainty quantification, stochastic parametrized advection diffusion equations, advection dominated problems.

No abstract

No abstract

This work introduces an extension of the residual distribution (RD) framework to stiff relaxation problems. The RD is a class of schemes which is used to solve a hyperbolic system of partial differential equations. To our knowledge, it has been used only for systems with mild source terms, such as gravitation problems or shallow water equations. What we propose is an implicit-explicit (IMEX) version of the RD schemes that can resolve stiff source terms, without refining the discretization up to the stiffness scale. This can be particularly useful in various models, where the stiffness is given by topological or physical quantities, e.g., multiphase flows, kinetic models, or viscoelasticity problems. We will focus on kinetic models that are BGK approximation of hyperbolic conservation laws. The extension to more complicated problems will be carried out in future works. The provided scheme is able to catch different relaxation scales automatically, without losing accuracy; we prove that the scheme is asymptotic preserving and this guarantees that, in the relaxation limit, we recast the expected macroscopic behavior. To get a high order accuracy, we use an IMEX time discretization combined with a deferred correction procedure, while naturally RD provides high order space discretization. Finally, we show some numerical tests in one and two dimensions for stiff systems of equations.