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

Dispersive and dissipative of two interesting numerical methods for treating convective transport are investigated: the dispersion-relation-preservation (DRP) scheme, proposed by Tarn and Webb, and the unified space-time α-ε method, developed by Chang. The space-time α-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, and its interaction with the stability bound and choice of parameters. Even though both methods are explicit in time, the appropriate ranges of the CFL number, ν, are different between them. For the DRP scheme, it is preferable if ν is close to 0.2 for short wave, and close to 0.1 for intermediate and long wave. With v less than but close to 1, matching between v and s is found to 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 slower with the space-time α-ε scheme, while for short waves, errors accumulate slower with the DRP scheme.