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

thesis

posted on 20.09.2018 by Arnab Debnath#### thesis

In order to distinguish essays and pre-prints from academic theses, we have a separate category. These are often much longer text based documents than a paper.

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).

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).