Eigensolution analysis of the discontinuous Galerkin method with nonuniform grids: I. one space dimension

  • Authors:

  • Fang Q. Hu;Harold L. Atkins

  • Affiliations:

  • Department of Mathematics and Statistics, Old Dominion University, Norfolk, Virginia;NASA Langley Research Center, Hamptom, Virginia

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2002

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Abstract

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 semidiscrete 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 differential 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 flux formula is used, the spurious mode becomes nonexistent. 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 effects 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 reflections at grid discontinuity, when they occur, are always in the form of the spurious nonphysical mode. Closed-form numerical reflection and transmission coefficients are given and analyzed. Numerical examples that illustrate the analytical findings of the paper are also presented.