Efficient numerical methods for the uncertain Boltzmann equation based on a hybrid solver

TL;DR

This paper introduces and compares advanced multi-level and multi-fidelity numerical methods for efficiently solving the uncertain Boltzmann equation, leveraging an asymptotic-preserving hybrid scheme to improve computational speed while preserving accuracy.

Contribution

It develops and evaluates APH-based MLMC and multi-fidelity methods for the Boltzmann equation with uncertainty, demonstrating their efficiency and practical guidelines for use.

Findings

Methods are significantly faster than standard approaches.

Both methods maintain physical properties and accuracy.

Guidelines help choose between MLMC and multi-fidelity based on problem characteristics.

Abstract

In this work, we propose and compare several approaches to solve the Boltzmann equation with uncertain parameters, including multi-level Monte Carlo and multi-fidelity methods that employ an asymptotic-preserving-hybrid (APH) scheme (Filbet and Rey, 2015) for the deterministic Boltzmann model. By constructing a hierarchy of models from finer to coarser meshes in phase space for the APH scheme and adopting variance reduction techniques, the MLMC method is able to allocate computational resources across different hierarchies quasi-optimally. On the other hand, in the bi-fidelity method we choose the APH scheme for the Boltzmann equation as the high-fidelity solver, and a finite volume scheme for the compressible Euler system as the low-fidelity model. Since both methods are non-intrusive, they can preserve the physical properties of the deterministic solver. Extensive numerical experiments demonstrate that our APH-based MLMC and multi-fidelity methods are significantly faster than standard approaches, while maintaining accuracy. We also provide practical guidelines for selection between APH-based MLMC and multi-fidelity approaches, based on solution smoothness and computational resource availability.