Capturing the Cross‐Terms in Multi‐Dimensional Advection Schemes

Solving the advection equation is an important part of numerically modelling the atmosphere. Both accuracy and efficiency are desirable traits of an advection scheme. For multi‐dimensional flow, forward‐in‐time advection schemes must properly capture the cross‐terms. Failure to capture the cross‐terms can result in reduced accuracy and even instabilities. We show how multi‐dimensional forward‐in‐time schemes successfully capture the cross‐terms of two‐dimensional flow. We then introduce a method to improve the efficiency of the forward‐in‐time schemes for two‐dimensional flow. This method stacks the duplicated cross‐terms from one flux into the other, creating asymmetrized fluxes. Numerical testing shows that these asymmetrized flux calculation schemes perform to the same accuracy as the original forwards‐in‐time schemes but with a significant improvement in computational time. Finally, we show extensions of the method to three‐dimensional flow.