Comparing schemes for integrating the Euler equations

Over the years there have been a number of studies comparing the relative merits of semi-Lagrangian and Eulerian schemes. These studies, which continue to appear in the literature up to the present, almost invariably conclude that semi-Lagrangian schemes are superior in accuracy, and produce less noise, than Eulerian schemes. It is argued in this note that such conclusions are not justified because they have compared semi-Lagrangian and Eulerian schemes of different orders of accuracy. Typically, the semi-Lagrangian schemes tested have employed cubic spatial interpolation (and therefore are third order) in space, whereas the Eulerian schemes have usually been second order (and sometimes fourth order) in space. It is shown here that when semi-Lagrangian and Eulerian schemes of the same order are applied to the test case, namely, that of 'warm bubble' convection, there are almost indiscernible differences between the simulations. The contention presented here, therefore, is that it is the order of the scheme that is of primary importance, not whether it is semi-Lagrangian or Eulerian.