Efficient Adaptive Stochastic Collocation Strategies for Advection-Diffusion Problems with Uncertain Inputs

TL;DR

This paper introduces an efficient adaptive stochastic collocation method for solving advection-diffusion PDEs with uncertain inputs, combining parameter space approximation with adaptive time-stepping for improved accuracy and efficiency.

Contribution

It presents a novel non-intrusive adaptive strategy that integrates stochastic collocation with adaptive timestepping, driven by hierarchical error estimators.

Findings

The method effectively balances parametric and temporal errors.

It improves accuracy in time-dependent uncertain PDEs.

The approach is non-intrusive and adaptable to complex models.

Abstract

Physical models with uncertain inputs are commonly represented as parametric partial differential equations (PDEs). That is, PDEs with inputs that are expressed as functions of parameters with an associated probability distribution. Developing efficient and accurate solution strategies that account for errors on the space, time and parameter domains simultaneously is highly challenging. Indeed, it is well known that standard polynomial-based approximations on the parameter domain can incur errors that grow in time. In this work, we focus on advection-diffusion problems with parameter-dependent wind fields. A novel adaptive solution strategy is proposed that allows users to combine stochastic collocation on the parameter domain with off-the-shelf adaptive timestepping algorithms with local error control. This is a non-intrusive strategy that builds a polynomial-based surrogate that is adapted sequentially in time. The algorithm is driven by a so-called hierarchical estimator for the parametric error and balances this against an estimate for the global timestepping error which is derived from a scaling argument.