A Higher-Order Conservation Element Solution Element Method for Solving Hyperbolic Differential Equations on Unstructured Meshes
Air Force Research Laboratory/RQRS Edwards AFB
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This dissertation presents an extension of the Conservation Element Solution Element CESE method from second- to higher-order accuracy. The new method retains the favorable characteristics of the original second-order CESE scheme, including i the use of the space-time integral equation for conservation laws, ii a compact mesh stencil, iii the scheme will remain stable up to a CFL number of unity, iv a fully explicit, time-marching integration scheme, v true multidimensionality without using directional splitting, and vi the ability to handle two- and three-dimensional geometries by using unstructured meshes. This algorithm has been thoroughly tested in one, two and three spatial dimensions and has been shown to obtain the desired order of accuracy for solving both linear and non-linear hyperbolic spatial differential equations. The scheme has also shown its ability to accurately resolve discontinuities in the solutions. Higher order unstructured methods such as the Discontinuous Galerkin method and the Special Volume methods have been developed for one, two- and three-dimensional application. Although these schemes have seen extensive development and use, certain drawbacks of these methods have been well documented. For example, the explicit versions of these two methods have very stringent stability criteria. This stability criteria requires that the time step be reduced as the order of the solver increases, for a given simulation on a given mesh.