A bi-fidelity stochastic collocation method for transport equations with diffusive scaling and multi-dimensional random inputs

TL;DR

This paper develops a bi-fidelity stochastic collocation method for efficiently solving linear transport equations with random inputs under diffusive scaling, combining high- and low-fidelity models to improve computational speed and accuracy.

Contribution

It extends bi-fidelity stochastic collocation to transport equations with diffusive scaling, using an asymptotic-preserving scheme and a two-velocity Goldstein-Taylor model for acceleration.

Findings

Method achieves significant speed-up in uncertainty quantification.

High accuracy maintained even in non-diffusive regimes.

Numerical experiments validate efficiency and accuracy.

Abstract

In this paper, we consider the development of efficient numerical methods for linear transport equations with random parameters and under the diffusive scaling. We extend to the present case the bi-fidelity stochastic collocation method introduced in [33,50,51]. For the high-fidelity transport model, the asymptotic-preserving scheme [29] is used for each stochastic sample. We employ the simple two-velocity Goldstein-Taylor equation as low-fidelity model to accelerate the convergence of the uncertainty quantification process. The choice is motivated by the fact that both models, high fidelity and low fidelity, share the same diffusion limit. Speed-up is achieved by proper selection of the collocation points and relative approximation of the high-fidelity solution. Extensive numerical experiments are conducted to show the efficiency and accuracy of the proposed method, even in non diffusive regimes, with empirical error bound estimations as studied in [16].