Computer Science Theses and Dissertations

Permanent URI for this collectionhttp://hdl.handle.net/1903/2756

Browse

Filters

2013 - 20182

Settings

Subject: search.filters.subject.Compact schemes

Search Results

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    Central Compact-Reconstruction WENO Methods
    (2018) Cooley, Kilian; Baeder, James D; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    High-order compact upwind schemes produce block-tridiagonal systems due to performing the reconstruction in the characteristic variables, which is necessary to avoid spurious oscillations. Consequently they are less efficient than their non-compact counterparts except on high-frequency features. Upwind schemes lead to many practical drawbacks as well, so it is desirable to have a compact scheme that is more computationally efficient at all wavenumbers that does not require a characteristic decomposition. This goal cannot be achieved by upwind schemes so we turn to the central schemes, which by design require neither a Riemann solver nor a determination of upwind directions by characteristic decomposition. In practice, however, central schemes of fifth or higher order apparently need the characteristic decomposition to fully avoid spurious oscillations. The literature provides no explanation for this fact that is entirely convincing; however, a comparison of two central WENO schemes suggests one. Pursuing that possibility leads to the first main contribution of this work, which is the development of a fifth-order, central compact-reconstruction WENO (CCRWENO) method. That method retains the key advantages of central and compact schemes but does not entirely avoid characteristic variables as was desired. The second major contribution is to establish that the role of characteristic variables is to to make flux Jacobians within a stencil more diagonally dominant, having ruled out some plausible alternative explanations. The CCRWENO method cannot inherently improve the diagonal dominance without compromising its key advantages, so some strategies are explored for modifying the CCRWENO solution to prevent the spurious oscillations. Directions for future investigation and improvement are proposed.
  • Thumbnail Image
    Item
    Compact-Reconstruction Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws
    (2013) Ghosh, Debojyoti; Baeder, James D; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    A new class of non-linear compact interpolation schemes is introduced in this dissertation that have a high spectral resolution and are non-oscillatory across discontinuities. The Compact-Reconstruction Weighted Essentially Non-Oscillatory (CRWENO) schemes use a solution-dependent combination of lower-order compact schemes to yield a high-order accurate, non-oscillatory scheme. Fifth-order accurate CRWENO schemes are constructed and their numerical properties are analyzed. These schemes have lower absolute errors and higher spectral resolution than the WENO scheme of the same order. The schemes are applied to scalar conservation laws and the Euler equations of fluid dynamics. The order of convergence and the higher accuracy of the CRWENO schemes are verified for smooth solutions. Significant improvements are observed in the resolution of discontinuities and extrema as well as the preservation of flow features over large convection distances. The computational cost of the CRWENO schemes is assessed and the reduced error in the solution outweighs the additional expense of the implicit scheme, thus resulting in higher numerical efficiency. This conclusion extends to the reconstruction of conserved and primitive variables for the Euler equations, but not to the characteristic-based reconstruction. Further improvements are observed in the accuracy and resolution of the schemes with alternative formulations for the non-linear weights. The CRWENO schemes are integrated into a structured, finite-volume Navier-Stokes solver and applied to problems of practical relevance. Steady and unsteady flows around airfoils are solved to validate the scheme for curvi-linear grids, as well as overset grids with relative motion. The steady flow around a three-dimensional wing and the unsteady flow around a full-scale rotor are solved. It is observed that though lower-order schemes suffice for the accurate prediction of aerodynamic forces, the CRWENO scheme yields improved resolution of near-blade and wake flow features, including boundary and shear layers, and shed vortices. The high spectral resolution, coupled with the non-oscillatory behavior, indicate their suitability for the direct numerical simulation of compressible turbulent flows. Canonical flow problems -- the decay of isotropic turbulence and the shock-turbulence interaction -- are solved. The CRWENO schemes show an improved resolution of the higher wavenumbers and the small-length-scale flow features that are characteristic of turbulent flows. Overall, the CRWENO schemes show significant improvements in resolving and preserving flow features over a large range of length scales due to the higher spectral resolution and lower dissipation and dispersion errors, compared to the WENO schemes. Thus, these schemes are a viable alternative for the numerical simulation of compressible, turbulent flows.