Abstract

H-theorem is one of the important properties of the Boltzmann equation, which states the non-decreasing property of the Gibbs entropy. In this work, we are interested in finding a numerical scheme of the Boltzmann equation that preserves this property. For the spatially homogeneous Boltzmann equation, we consider a modification of the Fourier spectral method from the perspective of discrete velocity method, to achieve a second-order velocity discretization with the structure of detailed balance. The method allows one to readily apply the FFT -based fast algorithms, and it preserves positivity of the distribution function due to the application of a positive preserving limiter. As for the temporal discretization, we adopt a simple entropy fix by a convex combination of the current numerical solution and the equilibrium state. Such an entropy fix can be generalized to a wider class of ODE systems. It is proven that the entropy fix has only infinitesimal influence on the numerical order of the original scheme, and in many circumstances, it can be shown that the scheme does not affect the numerical order. The work on the spatially inhomogeneous case is ongoing. We will present some preliminary results on the analysis of the discontinuous Galerkin method.