Issures in Discontinuous High-Order Methods: Broadband Wave Computation and Viscous Boundary Layer Resolution

Abstract

A new discontinuous formulation named Correction Procedure via Reconstruction (CPR) was developed by Huynh [49] in 1D, and extended to simplex and hybrid meshes by Wang & Gao [107] for conservation laws. As with all discontinuous methods such as the discontinuous Galerkin (DG), spectral volume (SV) and spectral difference (SD) methods, CPR method employs a piecewise discontinuous space. All of them can be unified under the CPR framework, which is relatively simple to implement especially for high-order elements. In this thesis, we deal with two issues: the efficient computation of broadband waves, and the proper resolution of a viscous boundary layer with the high-order CPR method. A hybrid discontinuous space including polynomial and Fourier bases is employed in the CPR formulation in order to compute broad-band waves. The polynomial bases are used to achieve a certain order of accuracy, while the Fourier bases are able to exactly resolve waves at a certain frequency. Free-parameters introduced in the Fourier bases are optimized in order to minimize both dispersion and dissipation errors by mimicking the dispersion-relationpreserving (DRP) method for a one-dimensional wave problem. For the one-dimensional wave problem, the dispersion and dissipation properties and the optimization procedure are investigated through a wave propagation analysis. The optimization procedure is verified with a wave propagation analysis. This optimization procedure is verified through a mesh resolution analysis, which gives the relation between the grid points-per-wavelength (PPW) and the wave propagation distance. Numerical tests have been performed to verify the wave propagation properties for the scalar advection equation. The two-dimensional wave behavior is investigated through a wave propagation