# A bi-fidelity method for the multiscale Boltzmann equation with random parameters

## TL;DR

This paper introduces a bi-fidelity stochastic collocation method for efficiently solving the multiscale Boltzmann equation with random parameters, combining low- and high-fidelity models to accurately approximate macroscopic quantities.

## Contribution

The paper develops a bi-fidelity approach that leverages a low-fidelity Euler model and a few high-fidelity Boltzmann simulations for efficient uncertainty quantification.

## Key findings

- The bi-fidelity method accurately captures macroscopic quantities.
- The approach reduces computational cost significantly.
- Numerical experiments confirm high efficiency and accuracy.

## Abstract

In this paper, we study the multiscale Boltzmann equation with multi-dimensional random parameters by a bi-fidelity stochastic collocation (SC) method developed in [A. Narayan, C. Gittelson and D. Xiu, SIAM J. Sci. Comput., 36 (2014); X. Zhu, A. Narayan and D. Xiu, SIAM J. Uncertain. Quantif., 2 (2014)]. By choosing the compressible Euler system as the low-fidelity model, we adapt the bi-fidelity SC method to combine computational efficiency of the low-fidelity model with high accuracy of the high-fidelity (Boltzmann) model. With only a small number of high-fidelity asymptotic-preserving solver runs for the Boltzmann equation, the bi-fidelity approximation can capture well the macroscopic quantities of the solution to the Boltzmann equation in the random space. A priori estimate on the accuracy between the high- and bi-fidelity solutions together with a convergence analysis is established. Finally, we present extensive numerical experiments to verify the efficiency and accuracy of our proposed method.