Abstract
I present four interconnected research projects that explore various computational methods for the solution of evolution equations in loop quantum cosmology. The scheme considered is a loop-quantized description of the Schwarzschild black hole interior. In this framework, the continuous formalism of classical relativity is replaced by a discrete relation over the minimal length and volume units. The first project presents the use of von-Neumann stability analysis to denote criteria for avoiding exponential growth of the iterative evolution of the discrete relation. These criteria are related to the physical characteristics of the problem being solved. When these criteria are accounted for, a computational solution that exhibits the expected behavior can be produced. The second project develops a collocation-inspired approach to the solution of these iterative schemes. The approach utilizes the superposition of a modified Fourier basis as an assumed solution to the variable separated form of the discrete relation. The new formalism is demonstrated to exhibit the same stability condition described before; and a new physical, context to this stability condition is shown in relation to an analog of the Courant-Friedrichs-Lewy condition. The third project extends the approach discussed in the previous project by expressing a basis-representation as a tensor-product over the two variable separated descriptions of the evolution equation. Permitting a means to simultaneously compute solutions on a grid by solving a linear system of equations. The nuances of the numerics are presented and accounted for, and the resultant method is shown to be numerically robust and conducive to high-performance computing.