[Paper Review] Numerical Simulation of Microflows using Hermite Spectral Methods

Zhicheng Hu, Zhenning Cai+1 more|arXiv (Cornell University)|Jul 17, 2018

Gas Dynamics and Kinetic Theory28 references4 citations

TL;DR

This paper presents a Hermite spectral method for solving the spatially inhomogeneous Boltzmann equation using a generalized approximate quadratic collision operator for arbitrary distribution functions. By introducing a fast transform for efficient collision operator discretization, the method achieves high accuracy and convergence with O(M₀⁹ + M⁴) computational complexity, demonstrating strong agreement with DSMC results in 1D and 2D microflow benchmarks.

ABSTRACT

We propose a Hermite spectral method for the spatially inhomogeneous Boltzmann equation. For the inverse-power-law model, we generalize an approximate quadratic collision operator defined in the normalized and dimensionless setting to an operator for arbitrary distribution functions. An efficient algorithm with a fast transform is introduced to discretize this new collision operator. The method is tested for one-dimensional benchmark microflow problems.

Motivation & Objective

• To develop a deterministic, high-order numerical solver for the spatially inhomogeneous Boltzmann equation in rarefied gas dynamics.
• To extend the Hermite spectral method beyond the normalized, spatially homogeneous case to general distribution functions.
• To design an efficient algorithm for the collision operator that preserves quadratic form for low-order moments while enabling fast computation.
• To validate the method on benchmark microflow problems, including 1D and 2D lid-driven cavity flows with varying Knudsen numbers.
• To achieve better accuracy and smoother results than linear models or DSMC in hydrodynamic and dynamic regimes.

Proposed method

• Proposes a Hermite spectral method based on orthogonal polynomial expansion using Maxwellian-weighted Hermite polynomials to represent the distribution function.
• Generalizes the approximate quadratic collision operator from [29] to arbitrary distribution functions, preserving quadratic form only for the first few moments.
• Introduces a fast transform algorithm to efficiently compute the collision operator, reducing computational complexity to O(M₀⁹ + M⁴).
• Uses explicit time stepping with adaptive time steps on structured grids (100×100) for 2D simulations, with M₀ = 10 and M = 25 or 35 for degrees of freedom.
• Implements boundary conditions with potential Gibbs phenomenon issues, suggesting future use of filters or improved boundary treatments.
• Employs a CPU-based implementation on Intel Xeon E5-2680 v4 with 28 threads, measuring total and per-step CPU times for performance evaluation.

Experimental results

Research questions

• RQ1 Can a Hermite spectral method with a generalized approximate collision operator achieve high accuracy and efficiency in solving the spatially inhomogeneous Boltzmann equation for microflows?
• RQ2 How does the proposed fast transform algorithm reduce the computational cost of the collision operator while preserving physical consistency?
• RQ3 To what extent does the method reproduce DSMC results in 1D and 2D microflow benchmarks such as the lid-driven cavity flow?
• RQ4 What is the trade-off between accuracy and computational cost when varying the number of Hermite modes (M) and the order of the collision approximation (M₀)?
• RQ5 How do numerical artifacts like Gibbs phenomenon affect the solution near boundaries, and what improvements can be made?

Key findings

• The Hermite spectral method with the generalized collision operator achieves good agreement with DSMC results for both Kn = 0.1 and Kn = 1.0 in the 2D lid-driven cavity flow.
• For Kn = 0.1, the simulation used M = 25 and M₀ = 10, resulting in 3,276 degrees of freedom and a total CPU time of 6.93 × 10⁶ seconds.
• For Kn = 1.0, M = 35 and M₀ = 10 yielded 8,436 degrees of freedom and a total CPU time of 1.07 × 10⁷ seconds.
• The CPU time per degree of freedom was 1.07 × 10⁻⁵ s for Kn = 0.1 and 5.30 × 10⁻⁶ s for Kn = 1.0, indicating favorable scalability.
• Discrepancies near boundaries were observed, likely due to Gibbs phenomenon from discontinuous distribution functions, suggesting need for filtering or improved boundary conditions.
• The method demonstrates O(M₀⁹ + M⁴) time complexity, making it computationally promising for complex multi-dimensional and polyatomic gas problems.