Spectral difference method for unstructured grids I: Basic formulation

A new, high-order, conservative, and efficient method for conservation laws on unstructured grids is 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. Placements of these points with various orders of accuracy are given for the triangular 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 verified numerically. Numerical solutions of plane electromagnetic waves incident on perfectly conducting circular cylinders are presented and compared with the exact solutions to demonstrate the capability of the method. Excellent agreement has been found. The method is much simpler than the discontinuous Galerkin and spectral volume methods for unstructured grids.