Transport Theory (Advanced)

Dr. Zhenning Cai

Takes place in the second half of the semester as 4+2, following Introduction to Transport Theory in the first half.

The lecture will start from the first week of June. The time will be the same as the first half Introduction to Transport Theory. There will be lectures on every Monday from 10:15 to 11:45 and Tuesday from 10:15 to 11:45 in Rogowski 328. Exercise sessions will be on every Wendesday from 08:30 to 10:00. The first lecture will be on June 2. Please contact the instructor if you are interested in the lecture but not free at this time.

The lecture is in English. Course materials and further information will be found in L2P.

The Boltzmann equation in the gas kinetic theory is one of the most fundamental models in the kinetic theory. When the gas is rarefied, classical models such as Euler or Navier-Stokes equations are no longer valid, people have to turn to the Boltzmann equation for an accurate description of fluid states. However, due to its high dimensionality and complex form of collision term, it is quite challenging to solve this model numerically. When developing numerical schemes for the Boltzmann equation, one must take into account not only the numerical efficiency, but also physical aspects such as conservation and H-theorem. This course introduces the most recent numerical schemes for solving the nonlinear Boltzmann equation. It will be exhibited how standard numerical methods such as finite volume method, Newton iteration and fast Fourier transform are used in solving a complex partial differential equation. The lecture will cover the following topics:

• Kinetic models in the gas kinetic theory
• Conservative numerical method for solving the BGK-type Boltzmann equation
• Fast spectral method for solving the Boltzmann equation
• Implicit numerical scheme for solving the Boltzmann equation

• Kun Xu, Direct Modeling for Computational Fluid Dynamics – construction and application of unified gas-kinetic schemes, World Scientific (2015)
• G. Dimarco, L. Pareschi, Numerical methods for kinetic equations, Acta Numerica, Cambridge University Press, 2014, pp. 369-520
• Selected papers