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