Numerical Simulation of Microflows using Hermite Spectral Methods

We propose a Hermite spectral method for the spatially inhomogeneous Boltzmann equation. For the inverse-power-law model, we generalize an approximate quadratic collision operator defined in the normalized and dimensionless setting to an operator for arbitrary distribution functions. An efficient algorithm with a fast transform is introduced to discretize this new collision operator. The method is tested for one-dimensional benchmark microflow problems

Burnett Spectral Method for High-Speed Rarefied Gas Flows

We introduce a numerical solver for the spatially inhomogeneous Boltzmann equation using the Burnett spectral method. The modelling and discretization of the collision operator are based on the previous work [Z. Cai, Y. Fan, and Y. Wang, Burnett spectral method for the spatially homogeneous Boltzmann equation, arXiv:1810.07804], which is the hybridization of the BGK operator for higher moments and the quadratic collision operator for lower moments. To ensure the preservation of the equilibrium state, we introduce an additional term to the discrete collision operator, which equals zero when the number of degrees of freedom tends to infinity. Compared with the previous work [Z. Hu, Z. Cai, and Y. Wang,Numerical simulation of microflows using Hermite spectral methods, arXiv:1807.06236], the computational cost is reduced by one order. Numerical experiments such as shock structure calculation and Fourier flows are carried out to show the efficiency and accuracy of our numerical method

Burnett Spectral Method for the Spatially Homogeneous Boltzmann Equation

We develop a spectral method for the spatially homogeneous Boltzmann equation using Burnett polynomials in the basis functions. Using the sparsity of the coefficients in the expansion of the collision term, the computational cost is reduced by one order of magnitude for general collision kernels and by two orders of magnitude for Maxwell molecules. The proposed method can couple seamlessly with the BGK-type modelling techniques to make future applications affordable. The implementation of the algorithm is discussed in detail, including a numerical scheme to compute all the coefficients accurately, and the design of the data structure to achieve high cache hit ratio. Numerical examples are provided to demonstrate the accuracy and efficiency of our methodComment: 22 pages, 15 picutre

Hermite spectral method for Fokker-Planck-Landau equation modeling collisional plasma

We propose an Hermite spectral method for the Fokker-Planck-Landau (FPL) equation. Both the distribution functions and the collision terms are approximated by series expansions of the Hermite functions. To handle the complexity of the quadratic FPL collision operator, a reduced collision model is built by adopting the quadratic collision operator for the lower-order terms and the diffusive Fokker-Planck operator for the higher-order terms in the Hermite expansion of the reduced collision operator. The numerical scheme is split into three steps according to the Strang splitting, where different expansion centers are employed for different numerical steps to take advantage of the Hermite functions. The standard normalized Hermite basis [36] is adopted during the convection and collision steps to utilize the precalculated coefficients of the quadratic collision terms, while the one constituted by the local macroscopic velocity and temperature is utilized for the acceleration step, by which the effect of the external force can be simplified to an ODE. Projections between different expansion centers are achieved by an algorithm proposed in [29]. Several numerical examples are studied to test and validate our new metho

Bridging hydrodynamics and kinetic theory: Challenge from shock structure problems

We survey a number of moment hierarchies and test their performances in computing one-dimensional shock structures. It is found that for high Mach numbers, the moment hierarchies are either difficult to implement or hard to converge, making these methods questionable for the simulation of high-nonequilibrium flows. By examining the convergence issue of Grad's moment methods, we propose a new moment hierarchy to bridge the hydrodynamic models and the kinetic equation. Numerical tests show that the method is capable of predicting shock structures with high Mach numbers accurately, and the results converge to the solution of the Boltzmann equation as the number of moments increases