This dissertation presents an extension of the Conservation Element Solution Element (CESE) method from second to higherorder accuracy. The new method retains the favorable characteristics of the original secondorder CESE scheme, including (i) the use of the spacetime 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, timemarching integration scheme, (v) true multidimensionality without using directional splitting, and (vi) the ability to handle two and threedimensional 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 nonlinear hyperbolic partial 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 (DG) method and the Spectral Volume (SV) methods have been developed for one, two and threedimensional 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.
The research presented in this dissertation builds upon the work of Chang, who developed a fourthorder CESE scheme to solve a scalar onedimensional hyperbolic partial differential equation. The completed research has resulted in two key deliverables. The first is a detailed derivation of a highorder CESE methods on unstructured meshes for solving the conservation laws in two and threedimensional spaces. The second is the code implementation of these numerical methods in a computer code. For code development, a onedimensional solver for the Euler equations was developed. This work is an extension of Chang's work on the fourthorder CESE method for solving a onedimensional scalar convection equation. A generic formulation for the nthorder CESE method, where n > 3, was derived. Indeed, numerical implementation of the scheme confirmed that the order of convergence was consistent with the order of the scheme. For the two and threedimensional solvers, SOLVCON was used as the basic framework for code implementation. A new solver kernel for the fourthorder CESE method has been developed and integrated into the framework provided by SOLVCON. The main part of SOLVCON, which deals with unstructured meshes and parallel computing, remains intact. The SOLVCON code for data transmission between computer nodes for High Performance Computing (HPC).
To validate and verify the newly developed highorder CESE algorithms, several one, two and threedimensional simulations where conducted. For the arbitrary order, onedimensional, CESE solver, three sets of governing equations were selected for simulation: (i) the linear convection equation, (ii) the linear acoustic equations, (iii) the nonlinear Euler equations. All three systems of equations were used to verify the order of convergence through mesh refinement. In addition the Euler equations were used to solve the ShuOsher and Blastwave problems. These two simulations demonstrated that the new highorder CESE methods can accurately resolve discontinuities in the flow field.
For the twodimensional, fourthorder CESE solver, the Euler equation was employed in four different test cases. The first case was used to verify the order of convergence through mesh refinement. The next three cases demonstrated the ability of the new solver to accurately resolve discontinuities in the flows. This was demonstrated through: (i) the interaction between acoustic waves and an entropy pulse, (ii) supersonic flow over a circular blunt body, (iii) supersonic flow over a guttered wedge.
To validate and verify the threedimensional, fourthorder CESE solver, two different simulations where selected. The first used the linear convection equations to demonstrate fourthorder convergence. The second used the Euler equations to simulate supersonic flow over a spherical body to demonstrate the scheme's ability to accurately resolve shocks. All test cases used are well known benchmark problems and as such, there are multiple sources available to validate the numerical results. Furthermore, the simulations showed that the highorder CESE solver was stable at a CFL number near unity.
