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Implementation and Validation of a New Spectral Difference Method for Hyperbolic Conservation Laws Using Raviart-Thomas Elements

Numerical schemes using locally discontinuous polynomial approximation are very popular for high order approximation of conservation laws. While the most widely used numerical schemes under this paradigm appears to be the Discontinuous Galerkin method, the Spectral Difference scheme has often been found attractive as well, because of its simplicity in formulation and implementation. Linear stability analysis studies showed that the scheme in its original form is not unconditionally linearly stable for triangular mesh elements. However, recently it has been shown that the scheme is linearly stable for triangles, if we use Raviart-Thomas polynomial space for flux interpolation. The present thesis work aimed at implementing this new variant of the Spectral Difference scheme in 2-dimensional domains to solve the linear advection equation and also the Euler equations, thereby proving its usability for more complex fluid flow problems. Full order of convergence was achieved for the linear advection problem. The Euler equations were solved around an airfoil for the subsonic flow cases. The flow features were captured well even with a coarse mesh. Linear stability analysis was performed and an optimal Strong Stability Preserving (SSP) time stepping scheme was analyzed with the new Spectral Difference discretization.