A Bi-fidelity numerical method for velocity discretization of Boltzmann equations

TL;DR

This paper presents a bi-fidelity numerical method for efficiently discretizing velocities in Boltzmann equations, combining low- and high-fidelity models to improve computational efficiency while maintaining accuracy across multiple scales.

Contribution

The paper introduces a novel bi-fidelity algorithm that uses a greedy approach to select velocity points, integrating asymptotic-preserving schemes for Boltzmann equations across different regimes.

Findings

Effective velocity discretization for Boltzmann equations demonstrated

Robustness across various regimes and initial conditions shown

Error bounds and asymptotic-preserving properties established

Abstract

In this paper, we introduce a bi-fidelity algorithm for velocity discretization of Boltzmann-type kinetic equations under multiple scales. The proposed method employs a simpler and computationally cheaper low-fidelity model to capture a small set of significant velocity points through the greedy approach, then evaluates the high-fidelity model only at these few velocity points and to reconstruct a bi-fidelity surrogate. This novel method integrates a simpler collision term of relaxation type in the low-fidelity model and an asymptotic-preserving scheme in the high-fidelity update step. Both linear Boltzmann under diffusive scaling and the nonlinear full Boltzmann in hyperbolic scaling are discussed. We show the weak asymptotic-preserving property and empirical error bound estimates. Extensive numerical experiments on linear semiconductor and nonlinear Boltzmann problems with smooth or discontinuous initial conditions and under various regimes have been carefully studied, which demonstrates the effectiveness and robustness of our proposed scheme.