Eigensolution Analysis of the Discontinuous Galerkin Method with Non-uniform Grids, Part I: One Space Dimension

We present a detailed study of spatially propagating waves in a discontinuous Galerkin scheme applied to a system of linear hyperbolic equations. We start with an eigensolution analysis of the semi-discrete system in one space dimension with uniform grids. It is found that, for any given order of the basis functions, there are at most two spatially propagating numerical wave modes for each physical wave of the Partial Di erential Equations (PDE). One of the modes can accurately represent the physical wave of the PDE and the other is spurious. The directions of propagation of these two numerical modes are opposite, and, in most practical cases, the spurious mode has a large damping rate. Furthermore, when an exact characteristics split ux formula is used, the spurious mode becomes non-existent. For the physically accurate mode, it is shown analytically that the numerical dispersion relation is accurate to order 2p+ 2 where p is the highest order of the basis polynomials. The results of eigensolution analysis are then utilized to study the e ects of a grid discontinuity, caused by an abrupt change in grid size, on the numerical solutions at either side of the interface. It is shown that, due to mode decoupling , numerical re ections at grid discontinuity, when they occur, are always in the form of the spurious non-physical mode. Closed form numerical re ection and transmission coe cients are given and analyzed. Numerical examples that illustrate the analytical ndings of the paper are also presented.