Parallel ENO schemes applied to shock/cylinder interactions and numerical methods by Radon transform

The construction of an implementation of the high-order ENO finite difference method for the computation of compressible reactive supersonic flow with real gas equations of state on the CM5 parallel supercomputer is described. The equation of state is given by p = {dollar}rho RT{dollar}, and a polynomial fit is used to describe {dollar}Csb{lcub}p{rcub}.{dollar} An empirical model is used for the viscosity parameter {dollar}mu.{dollar} The thermal and species diffusion rates are derived from this by use of Prandtl and Schmidt numbers respectively. A single step, non-stiff reaction provides a crude model for hydrogen/oxygen combustion.; The numerical method is applied to the two-dimensional calculation of planar shock waves interacting with cylindrical light gas domains contained in air. The interaction with a combusting hydrogen gas inclusion is also computed. The results are compared with existing experiments, previous computational results, and with a shock-capturing spectral method.; The results suggest that numerical diffusion may have seriously contaminated the results of previous researchers, and that mixing efficiencies are considerably less than those computed previously. On the other hand, high wave number perturbations of a sharp cylindrical interface are observed to cause intense mixing. This contradicts previous observations from lower resolution simulations.; The high order ENO scheme compares surprisingly well with the spectral method. The comparison is increasingly close as the formal order of accuracy of the scheme and/or spatial resolution is increased. The results confirm that high order methods are essential for accurate simulation of time-varying flow fields.; In the second part of the thesis a basis derived from a singular value decomposition of the Radon transform is used to construct spectral methods for a 2D circular disk. A numerical scheme is also created which consists of a finite difference method operating in Radon space. Both methods are applied to the computation of linear hyperbolic flows. The spectral method has no singularities at the origin, and is seen to perform well when tau boundary conditions are applied.