Numerical Solution of the Non-equilibrium Boltzmann Equation using the Discontinuous Galerkin Finite Element Method

In this thesis, we describe a deterministic method of solving a four-dimensional reduced form of
the spatially homogeneous non-equilibrium Boltzmann Equation from its original seven-dimensional
form. We have used Discontinuous Galerkin discretization to seek solution in the velocity space for
different kinds of affine and viscometric fluid flows given by the macroscopic Eulerian velocity field
v(x; t) = A(I + tA)􀀀1x [15]. The symmetry properties of the Collision operator, the uniformity of
our mesh and the construction of our nodal DG basis on Gauss-quadrature nodes have reduced the
calculation of the collision kernel to O(n5), as shown by Josyula et al.[3], which has made it possible
for us to look into non-equilibrium Boltzmann equation. In this method the collision operator is precomputed
and it is used to observe the evolution of the velocity distribution function for different kinds
of flows including Couette flow, incompressible vortex-like structures. The computation of the Collision
operator was parallelized using 351 processors with OpenMP API. The simulations run in this
work are based on spatially homogeneous hard-sphere potentials although this method is generalized
for any molecular potential. We have compared the predictions of all our simulations of the Boltzmann
Equation with non-equilibrium molecular dynamics (NEMD).