A conservative and entropic scheme for the Boltzmann equation
A/Prof. Zhen-Ning Cai
National University of Singapore

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. 

About the Speaker


2022-09-27 10:00 AM
Room: Tencent Meeting
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