Jeffrey R. Haack

A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit

Abstract We present the formulation of a conservative spectral method for the Boltzmann collision operator with anisotropic scattering cross-sections. The method is an extension of the conservative spectral method of Gamba and Tharkabhushanam [17,18] , which uses the weak form of the collision operator to represent the collisional term as a weighted convolution in Fourier space. The method is tested by computing the collision operator with a suitably cut-off angular cross section and comparing the results with the solution of the Landau equation. We analytically study the convergence rate of the Fourier transformed Boltzmann collision operator in the grazing collisions limit to the Fourier transformed Landau collision operator under the assumption of some regularity and decay conditions of the solution to the Boltzmann equation. Our results show that the angular singularity which corresponds to the Rutherford scattering cross section is the critical singularity for which a grazing collision limit exists for the Boltzmann operator. Additionally, we numerically study the differences between homogeneous solutions of the Boltzmann equation with the Rutherford scattering cross section and an artificial cross section, which give convergence to solutions of the Landau equation at different asymptotic rates. We numerically show the rate of the approximation as well as the consequences for the rate of entropy decay for homogeneous solutions of the Boltzmann equation and Landau equation.

A spectral-Lagrangian Boltzmann solver for a multi-energy level gas

In this paper a spectral-Lagrangian method is proposed for the full, non-linear Boltzmann equation for a multi-energy level gas typical of a hypersonic re-entry flow. Internal energy levels are treated as separate species and inelastic collisions (leading to internal energy excitation and relaxation) are accounted for. The formulation developed can also be used for the case of a gas mixture made of monatomic gases without internal energy (where only elastic collisions occur). The advantage of the spectral-Lagrangian method lies in the generality of the algorithm in use for the evaluation of the elastic and inelastic collision operators, as well as the conservation of mass, momentum and energy during collisions. The latter is realized through the solution of constrained optimization problems. The computational procedure is based on the Fourier transform of the partial elastic and inelastic collision operators and exploits the fact that these can be written as weighted convolutions in Fourier space with no restriction on the cross-section model. The feasibility of the proposed approach is demonstrated through numerical examples for both space homogeneous and in-homogeneous problems. Computational results are compared with those obtained by means of the DSMC method in order to assess the accuracy of the proposed spectral-Lagrangian method.

High performance computing with a conservative spectral Boltzmann solver

We present new results building on the conservative deterministic spectral method for the space inhomogeneous Boltzmann equation developed by Gamba and Tharkabhushaman. This approach is a two-step process that acts on the weak form of the Boltzmann equation, and uses the machinery of the Fourier transform to reformulate the collisional integral into a weighted convolution in Fourier space. A constrained optimization problem is solved to preserve the mass, momentum, and energy of the resulting distribution. We extend this method to second order accuracy in space and time, and explore how to leverage the structure of the collisional formulation for high performance computing environments. The locality in space of the collisional term provides a straightforward memory decomposition, and we perform some initial scaling tests on high performance computing resources. We also use the improved computational power of this method to investigate a boundary-layer generated shock problem that cannot be described by cl...

A Fast Spectral Method for the Boltzmann Collision Operator with General Collision Kernels

We propose a simple fast spectral method for the Boltzmann collision operator with general collision kernels. In contrast to the direct spectral method \cite{PR00, GT09} which requires

Spectral method for a kinetic swarming model

O(N^6)

Conservative deterministic spectral Boltzmann solver near the grazing collisions limit

memory to store precomputed weights and has

Interfacial mixing in high-energy-density matter with a multiphysics kinetic model

O(N^6)

Investigation of nonequilibrium internal energy excitation in shock waves by means of a spectral-Lagrangian Boltzmann solver

numerical complexity, the new method has complexity

A fast conservative spectral solver for the nonlinear Boltzmann collision operator

O(MN^4\log N)

Deterministic computation of the Boltzmann transport equation with anisotropic scattering cross section

, where