A Polynomial Spectral Calculus for Analysis of DG Spectral Element Methods

We introduce a polynomial spectral calculus that follows from the summation by parts property of the Legendre-Gauss-Lobatto quadrature. We use the calculus to simplify the analysis of two multidimensional discontinuous Galerkin spectral element approximations

Multi-Dimensional Spectral Difference Method for Unstructured Grids

A new, high-order, conservative, and efficient method for conservation laws on unstructured grids has been developed. It combines the best features of structured and unstructured grid methods to attain computational efficiency and geometric flexibility; it utilizes the concept of discontinuous and high-order local representations to achieve conservation and high accuracy; and it is based on the finite-difference formulation for simplicity. Universal reconstructions are obtained by distributing unknown and flux points in a geometrically similar manner for all unstructured cells. Placement of these points with various orders of accuracy are given for triangular and tetrahedral elements. Accuracy studies of the method are carried out with the two-dimensional linear wave equation and Burgers ’ equation, and each order of accuracy is numerically verified. Numerical solutions of plane electromagnetic waves incident on perfectly conducting circular cylinders and spheres are presented and compared with the exact solutions to demonstrate the capability of the method. Excellent agreements have been found. The method is simpler and more efficient than previous discontinuous Galerkin and spectral volume methods for unstructured grids. Nomenclature d = dimension of the domain F = generalized flux i = index for cell j = index for unknown u k = index for flux F l = index for face L j M k r j = cardinal basis or shape function for unknown u = cardinal basis or shape function for flux F = unknown location rk = flux locatio