Communications in Mathematical Sciences

Volume 18 (2020)

Number 4

Diagram notation for the derivation of hyperbolic moment systems

Pages: 1149 – 1177



Julian Koellermeier (Mathematical Sciences, Peking University, Beijing, China)

Yuwei Fan (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)


We propose a diagram notation for the derivation of hyperbolic moment models for the Boltzmann equation that yields a better understanding of the resulting moment systems. So far several hyperbolic moment models were presented, but their derivations are often very technical and there is little insight into the explicit form of the equations. In our diagram notation, each term in the moment equations can be explicitly tracked throughout the derivation process and whether the resulting moment system is hyperbolic can be easily observed from the diagram. We apply the diagram notation to derive existing moment models, including Grad’s moment equations, hyperbolic moment equations, quadrature-based moment equations, and a new set of simplified hyperbolic moment equations that was rarely studied before. The differences in the derivation are easily explained in the diagram notation and the explicit form of the equations can be computed straightforwardly as opposed to existing frameworks.


kinetic theory, moment method, hyperbolicity, Grad, diagram notation

2010 Mathematics Subject Classification

35L02, 35Q20, 76P05, 82C40

Received 2 February 2018

Accepted 28 January 2020

Published 28 July 2020