This dissertation discusses two aspects of computational fluid dynamics: high order spatial accuracy and convergence-rate acceleration through system preconditioning. Concerning high-order accuracy, the computational qualities of various spatial methods for the finite-volume solution of the Euler equations are presented. The two-dimensional essentially non-oscillatory (ENO), k-exact, and dimensionally split ENO reconstruction operators are discussed and compared in terms of reconstruction and solution accuracy and computational cost. Standard variable extrapolation methods are included for completeness. Inherent steady-state convergence difficulties are demonstrated for adaptive-stencil algorithms. Methods for reconstruction error analysis are presented and an exact solution to the heat equation is used as an example. Numerical experiments presented include the Ringleb flow for numerical accuracy and a shock-reflection problem. A vortex-shock interaction demonstrates the ability of the EN 0 scheme to excel in capturing unsteady high-frequency flow physics.

Concerning convergence-rate acceleration, characteristic-wave preconditioning is extended to include generalized finite-rate chemistry with non-equilibrium thermodynamics Additionally, the proper preconditioning for the one-dimensional Navier-Stokes equations is presented. Eigenvalue stiffness is resolved and convergencerate acceleration is demonstrated over the entire Mach-number range from the incompressible to the hypersonic. Specific benefits are realized at low and transonic flow speeds. The extended preconditioning matrix accounts for thermal and chemical non-equilibrium and its implementation is explained for both explicit and implicit time marching. The effects of high-order spatial accuracy and various flux splittings are investigated. Numerical analysis reveals the possible theoretical improvements from using preconditioning at all Mach numbers. Numerical results confirm the expectations from the analysis. The preconditioning matrix is applied with dual time stepping to obtain arbitrarily high-order accurate temporal solutions within an implicit formulation. Representative test cases include flows with previously troublesome embedded high-condition-number regions.