“…Most of them were proposed for the shallow water equations over a non-flat bottom topology, another prototype example of hyperbolic balance laws; see, e.g., [3,12,19,43,1,37,42,40] and the references therein. In recent years, well-balanced numerical methods for the Euler equations (1) with gravitation have been designed within several different frameworks, including but not limited to the finite volume methods [20,4,15,5,21,16,17,13], gas-kinetic schemes [44,25], finite difference methods [39,10,24], and finite element discontinuous Galerkin (DG) methods [22,6,23,30]. Recently, comparison between high-order DG method and well-balanced DG methods was carried out in [30].…”