This thesis is on numerical simulation of cyclic thermodynamic processes. A modelling approach and a method for finding periodic steady state solutions are described. Examples of applications are given in the form of four research papers. Stirling machines and pulse tube coolers are introduced and a brief overview of the current state of the art in methods for simulating such machines is presented. It was found that different simulation approaches, which model the machines with different levels of detail, currently coexist. Methods using many simplifications can be easy to use and can provide results quickly, but they are limited with respect to the phenomena that can be studied. More comprehensive methods can be used to study and optimise machines or components in more detail, but they usually require more time and computer resources. In this work the focus was on methods that are fast enough to be used for numerical optimisation of complete machine designs. The highest level of detail which appears to be feasible for this purpose is to model the gas flows in the machines as being primarily one-dimensional. The theory and implementation of a control volume based approach for modelling oscillating, compressible flow in one space dimension is presented. The implementation produces models where all the equations, which are on a form that should be understandable to someone with a background in engineering thermodynamics, can be accessed and modified individually. The implementation was designed to make models flexible and easy to modify, and to make simulations fast. A high level of accuracy was achieved for integrations of a model created using the modelling approach; the accuracy depended on the settings for the numerical solvers in a very predictable way. Selection of fast numerical algorithms and multi-threading accelerated simulations considerably. The discretisation scheme of the modelling approach was found to be convergent, and even relatively coarse discretisations produced useable results. Models created using the modelling approach produced results in good agreement with experimental data, and with simulation results from current state of the art software, for two Stirling machines and two pulse tube coolers. Parallelised single and multiple shooting methods were studied and were found to be reliable for finding periodic steady state solutions. Multiple shooting methods had better parallel scalability but this advantage was almost neutralised by a significant overhead compared to single shooting. The overhead was due to transients at the beginnings of the sub intervals in the cycle. The severity of the overhead was specific to models which included the inertia of the gas in the momentum balance. Single shooting, where the fastest evolving variables such as velocities and pressures, where excluded from the shooting, was the fastest sequential method. Fixed point iteration was performed on the excluded variables during the shooting. The parallel scalability for batch jobs was improved with an implementation which uses the parallelism inherent in batch jobs to increase the scalability of the parallelised shooting methods. The four research papers are self contained studies on: 1) the effects of regenerator matrix temperature oscillations on the performance of a Stirling engine, 2) optimal regenerator designs which takes into account the effects of the regenerator matrix temperature oscillations, 3) transverse asymmetry in the temperature profile of a regenerator in a pulse tube cooler, and 4) the appendix gap losses in a Stirling engine.