High-order methods for computational fluid dynamics

In the past two decades, the growing interest in the study of fluid flows involving discontinuities, such as shocks or high gradients, where a quadratic-convergent method may not provide a satisfactory solution, gave a notable impulse to the employment of high-order techniques. The present dissertation comprises the analysis and numerical testing of two high-order methods. The first one, belonging to the discontinuous finite-element class, is the discontinuous control-volume/finite-element method (DCVFEM) for the advection/ diffusion equation. The second method refers to the high-order finite-difference class, and is the mixed weighted non-oscillatory scheme (MWCS) for the solution of the compressible Euler equations. The methods are described from a formal point of view, a Fourier analysis is used to assess the dispersion and dissipation errors, and numerical simulations are conducted to confirm the theoretical results.