Efficient Numerical Methods for the Uncertain Boltzmann Equation Based on a Hybrid Solver

Abstract

Abstract. In this work, we propose and compare several approaches to solve the Boltzmann equation with uncertain parameters, including multilevel Monte Carlo and multifidelity 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 bifidelity 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 nonintrusive, they can preserve the physical properties of the deterministic solver. Extensive numerical experiments demonstrate that our APH-based MLMC and multifidelity methods are significantly faster than standard approaches, while maintaining accuracy. We also provide practical guidelines for selection between APH-based MLMC and multifidelity approaches based on solution smoothness and computational resource availability.