Spectral methods, thanks to the high accuracy and the possibility of using fast algorithms, represent an effective way to approximate collisional kinetic equations in kinetic theory. On the other hand, the loss of some local invariants can lead to the wrong long time behavior of the numerical solution. We introduce in this paper a novel Fourier-Galerkin spectral method that improves the classical spectral method by making it conservative on the moments of the approximated distribution, without sacrificing its spectral accuracy or the possibility of using fast algorithms. The method is derived directly using a constrained best approximation in the space of trigonometric polynomials and can be applied to a wide class of problems where preservation of moments is essential. We then apply the new spectral method to the evaluation of the Boltzmann collision term, and prove spectral consistency and stability of the resulting Fourier-Galerkin approximation scheme. Various numerical experiments illustrate the theoretical findings.

翻译：光谱方法,由于高度精度和使用快速算法的可能性,代表了在动能理论中近似碰撞动动方程的有效方法。 另一方面,某些局部变异物的丧失可能导致数字解决方案的错误长期行为。 我们在本文件中引入了一个新的Fourier-Galerkin光谱方法,通过在近似分布时使其保守,同时不牺牲光谱精确度或使用快速算法的可能性,改进了古典光谱方法。该方法是直接利用三角对称多角度空间中受限制的最佳近似法来得出的,可以适用于一系列广泛的问题,而这些问题对时间的保全至关重要。然后我们运用新的光谱方法来评估波尔茨曼相撞期,并证明由此形成的Fourier-Galerkin近似方法的光谱一致性和稳定性。各种数字实验都说明了理论结论。