| In this thesis we discuss two topics in structured and unstructured weighted essentially non-oscillatory (WENO) schemes. In the first part, we construct high order WENO schemes for solving the Hamilton-Jacobi equations on unstructured meshes. The main ideas here are nodal based approximations, the usage of monotone Hamiltonians as building blocks on unstructured meshes, nonlinear weights using smooth indicators of second and higher derivatives, and a strategy to choose diversified smaller stencils to make up the bigger stencil in the WENO procedure. Extensive numerical experiments are performed to demonstrate the stability and accuracy of the methods. High-order accuracy in smooth regions, good resolution of derivative singularities, and convergence to viscosity solutions are observed. The second part of this dissertation is a study in the resolution and numerical viscosities of high order finite difference WENO schemes. First, using the Euler equations of compressible flows, we compute solutions containing both discontinuities and complex solution features, through two representative numerical examples: the double Mach reflection problem and the Rayleigh-Taylor (RT) instability problem. We conclude that for solutions with such features, it is more economical in CPU time to use higher order WENO schemes than the lower order schemes to obtain comparable numerical resolution. Then the Navier-Stokes (NS) equations are considered. A quantitative study is carried out to investigate the size of numerical viscosities and the resolution of high order WENO schemes for solving NS equations for compressible flows at high Reynolds numbers. For problems with small scale structures like the RT instability problem, the details of the small structures are determined by the physical viscosity in the NS equations. Thus, in order to resolve these small scale structures, the inherent numerical viscosity of the scheme must be small enough so that the physical viscosity dominates. A mesh refinement study is performed to capture the threshold mesh for full resolution, when WENO schemes of different accuracy orders are used. It is demonstrated again that high order WENO schemes are more CPU time efficient than the lower order schemes to reach the same resolution, for our test problems. |
