Assessment of dispersion-relation-preserving and space-time CE/SE schemes for wave equations

Two interesting numerical methods for treating convective transport are investigated: the dispersion-relation-preservation (DRP) scheme, proposed by Tam and Webb, and the unified space-time a-ε method, developed by Chang. The space-time a-ε method directly controls the level of dispersion and dissipation via a free parameter, ε, while the DRP scheme minimizes the error by matching the characteristics of the wave. Insight into the dispersive and dissipative aspects in each scheme is gained from analyzing the truncation error. Even though both methods are explicit in time, the appropriate ranges of the CFL number, v, are different between them. For the DRP scheme, it is preferable if v is close to 0.2 for short waves, and close to 0.1 for intermediate and long waves. With v less than but close to 1, matching between v and ε can substantially affect the accuracy of the space-time method. For both methods, different performance characteristics are observed between long and short waves. It seems that for long waves, errors grow more slowly with the space-time a-ε scheme, while for short waves, errors accumulate more slowly with the DRP scheme.