An adaptive Hermite spectral method for the Boltzmann equation

We propose an adaptive Hermite spectral method for the three-dimensional velocity space of the Boltzmann equation guided by a newly developed frequency indicator. For the homogeneous problem, the indicator is defined by the contribution of high-order coefficients in the spectral expansion. For the non-homogeneous problem, a Fourier-Hermite scheme is employed, with the corresponding frequency indicator formulated based on distributions across the entire spatial domain. The adaptive Hermite method includes scaling and p-adaptive techniques to dynamically adjust the scaling factor and expansion order according to the indicator. Numerical experiments cover both homogeneous and non-homogeneous problems in up to three spatial dimensions. Results demonstrate that the scaling adaptive method substantially reduces L2 errors at negligible computational cost, and the p-adaptive method achieves time savings of up to 74%.