Issues in discontinuous high-order methods: Broadband wave computation and viscous boundary layer resolution

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 analysis too. The wave propagation properties are verified with a numerical test of the twodimensional acoustic wave equation.;In order to understand the mesh size requirement to resolve a viscous boundary layer using high-order methods, extensive grid resolution studies are performed for both 1D and 2D viscous burger's equations with exact solutions. The skin friction is used as an indicator of accuracy for the resolution of a boundary layer. For the diffusion terms, the local discontinuous Galerkin (LDG) method is employed to achieve the (k + 1)th order of accuracy with a degree k polynomial reconstruction.;For the 1D viscous burger's equation, different grid sizes are determined for various order CPR formulations given a certain error in the skin friction. And different skin frictions are obtained for a certain grid size. In addition, accuracy and convergence properties are studied for different distribution of solution points.;A 2D viscous burger's equation with an exact solution is designed to test the resolution for various orders of CPR formulations. The method of manufactured solution (MMS) is employed to provide an exact solution for code accuracy verification. In MMS, instead of solving the original equation directly, the equation with an analytical source term is solved. Accuracy studies are also carried out.;Keywords: (Correction Procedure via Reconstruction), A Hybrid Discontinuous Space, Wave Propagation Analysis, Grid Resolution Study, Method of Manufactured Solution.