A CRITICAL STUDY OF HIGHER ORDER DISCONTINUOUS FINITE ELEMENT METHODS FOR SOLUTION OF EULER EQUATIONS

Abstract

This thesis presents a critical study for higher order discontinuous finite element methods. This study includes flux reconstruction approach, which includes discontinuous Galerkin method and spectral difference method. The study is conducted in the light of Von Neumann stability analysis. Hence, two-dimensional solver for quadrilateral grid has been developed. Then, a criticism of the aforementioned method is presented based on Von Neumann analysis. This criticism shows that the utilization of polynomial based approximation does not always yield the well-established order of accuracy in literature. Also, it shows that Euler model is second order accurate as a consequence of modelling error. Hence, the utilization of higher order accurate numerical methods does not make sense in solving the Euler equations. Finally, a new development for finite difference method is proposed. This development enables us to get a second order accurate solution without seeking numerical boundary conditions.