A high‐order WENO‐limited finite‐volume algorithm for atmospheric flow using the ADER‐differential transform time discretization

A high‐order‐accurate weighted essentially non‐oscillatory (WENO) limited upwind finite‐volume scheme is detailed for the compressible, nonhydrostatic, inviscid Euler equations using an arbitrary derivatives (ADER) time‐stepping scheme based on differential transforms (DTs). A second‐order‐accurate alternating Strang dimensional splitting is compared against multidimensional simulation with 2D transport using solid body rotation of various data. The two were found to give nearly identical accuracy in orthogonal, Cartesian coordinates. Orders of convergence are demonstrated at up to ninth‐order accuracy with 2D transport. 1D transport is used to confirm that error decreases monotonically with increasing order of accuracy with WENO limiting even for discontinuous data. Further, WENO limiting always decreased the error compared with simulation without limiting in the L1 norm. A series of standard 2D compressible nonhydrostatic Euler equation test cases were validated against previous results from literature. Finally, it was demonstrated that increasing the order of accuracy led to better resolved features and increased power for kinetic energy at small wavelengths. A high‐order WENO‐limited finite‐volume algorithm for atmospheric flow using ADER‐differential transform time stepping is presented. The scheme is validated against literature standard test cases. The scheme is shown to be stable without WENO limiting or any explicit damping even at ninth‐order accuracy due to upwind fluxes. The effects of WENO limiting are demonstrated, and the increased resolution of higher‐order accuracy is shown.