High order operator splitting for the Boltzmann equation within the interaction picture

Abstract

Project aims to improve on a feature of the C++ library TORTOISE, the full time propagation of the Boltzmann equation comprising of the transport and scattering terms. This project focuses on the transport term and strives to reduce the cost and error by performing time propagation in the interaction picture. Obtaining an analytical solution to the Boltzmann equation is extremely difficult, and numerical methods are often used to approximate a solution. First half of the project aims to use the discontinuous Garlerkin method to discretise the spatial and velocity spaces into individual elements as an approximation of the density of particles. This approximation allows for easier computation, but also introduces jump discontinuity at the edges. Luckily, we know that the Dirac delta function is defined when integrated over its domain. Hence, the integral of the Boltzmann transport equation is also defined, and can be solved with integration by parts. Results of the integration tells us that it can be interpreted as a sum of the volume flux within the element and the surface flux at the edges. Thus, this project first involves the building of the explicit volume and surface flux matrices, then they are propagated with the Dormand-Prince Runge-Kutta method implemented in the interaction picture. The main issue with performing the propagation using conventional Dormand-Prince is that the transport operator requires a significantly smaller step size than the scattering operator to be stable. Implementation in the interaction picture will resolve this problem and we will observe that this will help to reduce the overall cost.