[Paper Review] A new stability and convergence proof of the Fourier-Galerkin spectral method for the spatially homogeneous Boltzmann equation

Jingwei Hu, Kunlun Qi+1 more|arXiv (Cornell University)|Jul 10, 2020

Gas Dynamics and Kinetic Theory21 references5 citations

TL;DR

This paper presents a new stability and convergence proof for the Fourier-Galerkin spectral method applied to the spatially homogeneous Boltzmann equation, using an $L^2$ estimate of the negative part of the solution rather than relying on the positivity or 'spreading' properties of the collision operator. The method ensures stability for both continuous and discontinuous initial data, offering a simpler and more general convergence framework than prior approaches.

ABSTRACT

Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier-Galerkin spectral method has become a popular deterministic method for solving the Boltzmann equation, manifested by its high accuracy and potential of being further accelerated by the fast Fourier transform. Albeit its practical success, the stability of the method is only recently proved by Filbet, F. & Mouhot, C. in [$ Trans. Amer. Math. Soc.$ 363, no. 4 (2011): 1947-1980.] by utilizing the "spreading" property of the collision operator. In this work, we provide a new proof based on a careful $L^2$ estimate of the negative part of the solution. We also discuss the applicability of the result to various initial data, including both continuous and discontinuous functions.

Motivation & Objective

• To establish a new, simpler stability proof for the Fourier-Galerkin spectral method applied to the spatially homogeneous Boltzmann equation.
• To remove reliance on the 'spreading' or positivity properties of the collision operator used in prior proofs.
• To quantify the initial data conditions under which the method remains stable, including for discontinuous functions.
• To provide a rigorous convergence proof with spectral accuracy under minimal assumptions on initial data.
• To offer a more general and accessible theoretical foundation for the method's practical success in kinetic simulations.

Proposed method

• The method employs an $L^2$ norm estimate of the negative part of the solution to control instability arising from spectral approximation.
• It avoids the need to enforce global positivity by showing that small initial negative parts remain controllable over time.
• The analysis is conducted on a torus via domain truncation and periodization, with careful treatment of the truncated collision operator.
• Key estimates are derived for the gain part of the collision operator using Riesz-Thorin interpolation and $L^p$ boundedness of the associated integral operator.
• The proof leverages the translation-invariance of the collision operator and the convolution-like structure enabled by Fourier basis projection.
• A priori bounds are established in $L^2$ and $L^p$ spaces to ensure local existence, uniqueness, and global well-posedness on bounded time intervals.

Experimental results

Research questions

• RQ1 Can the stability of the Fourier-Galerkin spectral method be proven without relying on the positivity or spreading properties of the collision operator?
• RQ2 What conditions on the initial data—continuous or discontinuous—ensure stability of the spectral method?
• RQ3 How can the $L^2$ norm of the negative part of the solution be controlled over time to guarantee stability?
• RQ4 What is the precise relationship between the initial data size and the stability threshold of the method?
• RQ5 Can spectral accuracy and convergence be rigorously established under minimal regularity assumptions on the initial data?

Key findings

• The method achieves stability for initial data with small $L^2$ norm of the negative part, even if the solution is not globally positive.
• The proof establishes local existence and uniqueness of the solution in $L^2$ for the truncated Fourier-Galerkin system.
• Global well-posedness and stability are proven on arbitrary bounded time intervals under the condition that the initial negative part is sufficiently small in $L^2$.
• The convergence rate of the method is spectral, as expected, and the proof is valid for both continuous and discontinuous initial data.
• The approach avoids the need for positivity-preserving filters, preserving spectral accuracy while maintaining stability.
• The result provides a simpler and more general theoretical framework than prior proofs relying on the 'spreading' property of the collision operator.