  In low-mass X-ray binaries, the accretion of stellar material onto a neutron
star can fuel unstable thermonuclear flashes known as Type I X-ray bursts.
Simulating these events using computational models can provide valuable
information about the nature of the accreting system. One-dimensional (1D)
astrophysics codes with large nuclear reaction networks are the current
state-of-the-art for simulating X-ray bursts. These codes can track the
evolution of isotopes through thousands of nuclear reaction pathways, to
predict the released nuclear energy and final composition of the ashes. In this
thesis, I make extensive use of KEPLER, a 1D code at the forefront of these
efforts. I first present improvements to the setup and analysis of KEPLER burst
models. By accounting for nuclear heating in the initial conditions, I shorten
the thermal burn-in time, thereby reducing computational expense and producing
more consistent burst trains. To model bursts fueled by transient accretion
events, I perform the first such simulations with fully time-dependent
accretion rates. Building upon previous efforts to model the "Clocked Burster",
GS 1826$-$238, I precompute a grid of 3840 simulations and sample the
interpolated results using Markov Chain Monte Carlo (MCMC) methods. By
comparing the predictions to multi-epoch observations, I obtain posterior
probability distributions for the system parameters. I then extend these MCMC
methods to the pure-helium burster, 4U 1820$-$30, using a grid of 168
simulations. Finally, I discuss potential improvements for future studies, to
further develop the computational modelling of accreting neutron stars.
