H-theorem is one of the important properties of the Boltzmann equation, which states the non-decreasing property of the Gibbs entropy. Meanwhile, it conserves the mass, momentum and energy, which are also fundamental laws in classical mechanics. In this work, we are interested in finding a numerical scheme of the Boltzmann equation that preserves both the entropy dissipation and the conservation laws. To achieve this, we first study a general ODE system with Gibbs entropy, and develop a simple entropy fix for entropy-violating solutions by a convex combination of the current numerical solution and the equilibrium state. It is shown rigorously that the entropy fix does not affect the numerical order. This approach can be applied to the Boltzmann equation if the numerical solution is positive. To this aim, we develop a positive-conservative projection method based on the spectral method for the Boltzmann equation. By combining the projection and the entropy fix, we obtain a numerical scheme with all desired properties. Numerical tests show that the scheme has better accuracy when the solution is close to the equilibrium. 

蔡振宁，新加坡国立大学助理教授。2008年于北京大学获理学学士学位，2013年于北京大学获计算数学方向博士学位。2014年至2015年获德国洪堡基金会资助于亚琛工业大学做博士后研究，2016年先后于美国马里兰大学和杜克大学担任访问助理教授。主要研究方向包括气体动理学的建模与计算、开放量子系统的模拟以及量子色动力学中复Langevin方法的理论研究，在国际学术期刊发表论文四十余篇。