Approximation of the Boltzmann collision operator based on hermite spectral method

Abstract

Based on the Hermite expansion of the distribution function, we introduce a Galerkin spectral method for the spatially homogeneous Boltzmann equation with the realistic inversepower-law models. A practical algorithm is proposed to evaluate the coefficients in the spectral method with high accuracy, and these coefficients are also used to construct new computationally affordable collision models. Numerical experiments show that our method captures the low-order moments very efficiently.

Burnett spectral method for the spatially homogeneous Boltzmann equation

“…However, the time complexity O(M 8 ) (or O(M 7 ) for Maxwell molecules) still gives huge computational cost when M is large, especially when solving spatially inhomogeneous problems. To make the computation even cheaper, we adopt the idea in [8,25] which replaces the right-hand side of (2.5a) byQ * lmn (t), defined as…”

“…Other methods include the fast discrete velocity method [21] and the discontinuous Galerkin method [1]. Another type of spectral method based on global orthogonal polynomials is also being studied recently [8,13,25]. In this paper, we follow the work [8,13] and adopt the spectral method based on Burnett polynomials [6], which has been applied to the linearized Boltzmann equation [8,9], and shows great potential to achieve higher numerical efficiency.…”

“…Therefore our numerical method can represent this steady-state solution exactly. In [25,17], a similar method using Hermite polynomials, which are also orthogonal polynomials associated with the weight function M(v), is studied. In principle, the spectral methods using Burnett and Hermite polynomials are essentially equivalent, especially when the series is truncated up to the same degree.…”

“…Such an advantage has been utilized in [25], where the explicit expressions of all the coefficients in the discretization are formulated using these symmetries. However, the superiority of the Burnett polynomials introduced in [6] is the fact that they are eigenfunctions of the linearized collision integral for Maxwell molecules [26].…”

“…In this paper, we are going to focus on the detailed implementation of the algorithm, including a much more accurate way to compute the coefficients, a detailed analysis of the computational cost, and the design of the data structure to achieve high computational efficiency. Meanwhile, we also emphasize the modelling technique introduced in [25] which allows flexible balancing between computational cost and modelling error.…”