Stabilized weighted reduced basis methods for parametrized advection dominated problems with random inputs

TL;DR

This paper introduces stabilized weighted reduced basis methods for efficiently solving parametrized advection-dominated problems with random inputs, combining stabilization techniques with reduced basis approaches to improve accuracy and computational efficiency.

Contribution

It develops a novel integration of stabilized reduced basis methods with weighted reduced basis techniques for stochastic problems, including a selective online stabilization strategy.

Findings

Effective reduction of computational costs

High accuracy maintained with online stabilization

Successful application to heat transfer problems

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.