A positive and moment-preserving Fourier spectral method

TL;DR

This paper introduces a Fourier spectral method that preserves positivity and moments, ensuring high accuracy and efficiency, with applications demonstrated in solving the Boltzmann equation.

Contribution

A novel Fourier spectral method with optimization-based positivity and moment preservation, featuring an efficient Newton solver and spectral accuracy analysis.

Findings

Method maintains spectral accuracy.

Newton solver exhibits quadratic convergence.

Numerical examples confirm high accuracy and effectiveness in Boltzmann equation applications.

Abstract

This paper presents a novel Fourier spectral method that utilizes optimization techniques to ensure the positivity and conservation of moments in the space of trigonometric polynomials. We rigorously analyze the accuracy of the new method and prove that it maintains spectral accuracy. To solve the optimization problem, we propose an efficient Newton solver that has quadratic convergence rate. Numerical examples are provided to demonstrate the high accuracy of the proposed method. Our method is also integrated into the spectral solver of the Boltzmann equation, showing the benefit of our approach in applications.