A stochastic Galerkin method with adaptive time-stepping for the Navier-Stokes equations

TL;DR

This paper develops an adaptive time-stepping stochastic Galerkin method for solving the time-dependent Navier-Stokes equations with random viscosity, demonstrating its efficiency and accuracy through numerical experiments.

Contribution

It extends the stochastic Galerkin framework to Navier-Stokes equations with random viscosity and proposes efficient solution strategies for the resulting linear systems.

Findings

The stochastic Galerkin method accurately captures the effects of random viscosity.

The proposed preconditioned Krylov solver improves computational efficiency.

Numerical experiments validate the method's effectiveness.

Abstract

We study the time-dependent Navier-Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion, and we use the stochastic Galerkin method to extend the methodology from [D. A. Kay et al., \textit{SIAM J. Sci. Comput.} 32(1), pp. 111--128, 2010] into this framework. For the resulting stochastic problem, we explore the properties of the resulting stochastic solutions, and we also compare the results with that of Monte Carlo and stochastic collocation. Since the time-stepping scheme is fully implicit, we also propose strategies for efficient solution of the stochastic Galerkin linear systems using a preconditioned Krylov subspace method. The effectiveness of the stochastic Galerkin method is illustrated by numerical experiments.