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.

翻译：本文提出了一种新颖的傅里叶谱方法，该方法利用优化技术确保三角多项式空间中的正定性及矩守恒特性。我们严格分析了新方法的精度，并证明其保持谱精度。为求解该优化问题，我们提出了一种具有二次收敛速率的高效牛顿求解器。数值算例验证了所提方法的高精度特性。此外，该方法已被集成至玻尔兹曼方程的谱求解器中，展示了其在应用中的优势。