A comparison of high-order explicit and non-oscillatory finite difference advection schemes for climate and weather models

Summary The Euler equations govern the behavior of a fluid in motion. They have long been used as a test-bed for assessing the accuracy and efficiency of numerical schemes for solving them. This study focused on advection dominated flows so all other terms in the equations are omitted. The study examined only explicit schemes and does not address the many alternative approaches such as semi-Lagrangian and implicit formulations. The schemes examined here were applied to test cases of increasing complexity as it is well-known that a particular scheme may work very well on some test problems but fail on others. To avoid such problems, we began with the standard tests such as the advection of sine waves and various other one-dimensional shapes, then moved on to two-dimensional problems and finally tested cases that have many scales.Summarizing, the results from the range of test cases, it was found that the schemes that perform best are high-order upwind schemes, with diminishing returns above fifth- and sixth-order.Finally, the schemes were applied to cases where scalar advection requires that constraints such as positive-definiteness be satisfied. For these problems, the Flux Corrected Transport (FCT) and weighted essentially non-oscillatory (WENO) schemes were also applied. Similar conclusions were drawn, namely, that fifth- and sixth-order FCT and WENO schemes produced excellent simulations. However, these schemes were more computationally expensive than the standard high-order upwind and centred schemes.