Many computational fluid dynamics codes use Riemann solvers on an unstaggered grid for finite volume methods, but this approach is computationally expensive compared to existing atmospheric dynamical cores equipped with hyper‐diffusion or other similar relatively simple diffusion forms. We present a simplified Low Mach number Approximate Riemann Solver (LMARS), made computationally efficient through assumptions appropriate for atmospheric flows: low Mach number, weak discontinuities, and locally uniform sound speed. This work will examine the dissipative and dispersive properties of LMARS using Von Neumann linearized analysis to the one‐dimensional linearized shallow water equations. We extend these analyses to higher‐order methods by numerically solving the Fourier‐transformed equations. It is found that the pros and cons due to grid staggering choices diminish with high‐order schemes. The linearized analysis is limited to modal, smooth solutions using simple numerical schemes, and cannot analyze solutions with discontinuities. To address this problem, this work presents a new idealized test of a discontinuous wave packet, a single Fourier mode modulated by a discontinuous square wave. The experiments include studies of well‐resolved and (near) grid‐scale wave profiles, as well as the representation of discontinuous features and the results are validated against the Von Neumann analysis. We find the higher‐order LMARS produces much less numerical noise than do inviscid unstaggered and especially staggered schemes while retaining accuracy for better‐resolved modes.