Numerical Assessment of Linear and Nonlinear Models for Compressible Flows

This paper presents a numerical comparison between linearized and nonlinear models for compressible fluid flow in several configurations: a Cauchy-type setting, a half-space model with prescribed free-surface velocity, a bounded-domain model, and a pressure-dependent compressibility model. The purpose is to assess, under identical initial and boundary data, when the linear approximation reproduces the pressure response of the nonlinear formulation and when nonlinear effects become significant. The governing equations are written in terms of the logarithmic density variable σ = ln(ρ/ρ0) and the velocity field. A three-dimensional Lax–Wendroff finite-difference discretization is used for the computations. The revision clarifies the computational setup, including the grid 31 × 31 × 21, the dimensional time step Δt = 600 s, the representative grid spacing Δx = 6667 m, and the corresponding Courant number 0.045 based on a maximum velocity magnitude of 0.5 m/s. The initial temperature and pressure ranges used in the numerical setup are also stated explicitly. The results indicate that the linear and nonlinear models are generally close in the interior of the computational domain for the reported parameter regimes, while larger discrepancies occur near boundaries and grow in time. The pressure-dependent compressibility case exhibits the strongest mismatch, which is consistent with the stronger nonlinear pressure-gradient scaling in that model. The paper also states the limitations of the present computations: density is used locally as a diagnostic variable in the pressure update, total mass is not tracked as a separate diagnostic, and no grid-refinement study was performed in the reported simulations.