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Explore a comprehensive lecture on deterministic solutions to the Boltzmann equation using fast spectral methods. Delve into the challenges of numerically solving this integro-differential equation governing fluid flow behavior across various physical conditions. Learn about a novel fast Fourier spectral method for the Boltzmann collision operator, leveraging its convolutional and low-rank structure. Discover how this framework applies to arbitrary collision kernels, multiple species, and inelastic collisions. Examine the coupling of the fast spectral method with discontinuous Galerkin discretization to create a highly accurate deterministic solver (DGFS) for the full Boltzmann equation. Compare results from standard benchmark tests, including rarefied Fourier heat transfer, Couette flow, and thermally driven cavity flow, against direct simulation Monte Carlo (DSMC) solutions. Gain insights into the Boltzmann equation's properties, numerical issues, and various approximation techniques, including spherical representation and simplification methods.
