Characteristic-Based Schemes for Dispersive Waves I. The Method of Characteristics for Smooth Solutions

In order to embark on the development of numerical schemes for stiff problems, we have studied a model of relaxing heat flow. To isolate those errors unavoidably associated with discretization, a method of characteristics is developed, containing three free parameters depending on the stiffness ratio. It is shown that such “decoupled” schemes do not take into account the interaction between the wave families and hence result in incorrect wave speeds. We also demonstrate that schemes can differ by up to two orders of magnitude in their rms errors even while maintaining secondorder accuracy. We show that n o method of characteristics solution can be better than second-order accurate. Next, we develop “coupled” schemes which account for the interactions, and here we obtain two additional free parameters, We demonstrate how coupling of the two wave families can be introduced in simple ways and how the results arc greatly enhanced by this coupling. Finally, numerical results for several decoupled and coupled schemes are presented, and we observe that dispersion relationships can be a very useful qualitative tool for analysis of numerical algorithms for dispersive waves.