Fluid models are widely used in macroscopic simulations of low-temperature gas discharges because they can describe plasma density, sheath motion, electron heating, ionization, and current response at a relatively low computational cost. However, in radio-frequency capacitively coupled plasmas (RF-CCPs), steep gradients in electron density and electric field, together with strong variations in charged-particle fluxes, occur near the oscillating sheath edges. Therefore, a numerically converged periodic solution does not necessarily guarantee physically reliable predictions. This work investigates how spatial resolution and interfacial flux discretization affect the predicted discharge behavior in a one-dimensional 13.56 MHz argon RF-CCP. A drift-diffusion fluid model based on the finite-volume method and a staggered grid is employed. The same governing equations, boundary conditions, transport and reaction data, and time-advancement strategy are used in all simulations so that the observed differences can be attributed to the spatial discretization of charged-particle transport. Two representative discharge conditions are considered: 1 Torr/40 V and 2 Torr/200 V. Four commonly used interfacial flux treatments are compared, namely the Scharfetter–Gummel exponential flux, the first-order upwind flux, the central-difference flux, and the van Leer total-variation-diminishing flux-limiter scheme. The analysis focuses on their physical consequences for electron flux, sheath thickness, electron power absorption, mean electron temperature, ionization source term, electron density, and current density. The results show that, under coarse-grid conditions, the influence of the interfacial flux treatment is first manifested in the electron-flux distribution. Through the coupling among the electron continuity equation, the Poisson equation, and the electron energy equation, flux-discretization errors produce correlated deviations in electron transport, space charge and electric field, sheath structure, electron power absorption, mean electron temperature, ionization source term, and electron density.Different flux treatments introduce distinct systematic biases into the predicted physical quantities. The first-order upwind flux introduces strong numerical dissipation, which weakens electron transport near the sheath edges and leads to underestimation of electron power absorption, ionization source term, and electron density. The central-difference flux is prone to nonphysical oscillations in steep-gradient sheath regions; in the 1 Torr/40 V case, it produces artificial peaks in the mean electron temperature and ionization source term, whereas in the 2 Torr/200 V case, the simulation fails to reach a stable periodic state. The Scharfetter–Gummel flux performs reasonably well on the coarse grid in the 1 Torr/40 V case, but in the 2 Torr/200 V case, it overestimates the electron flux and electron density, shifts the sheath boundary toward the electrode, and reduces the cycle-averaged sheath thickness by approximately 40.1\% relative to the reference solution. This sheath compression further enhances near-electrode electron power absorption and local ionization. The van Leer TVD flux exhibits comparatively robust behavior by suppressing central-difference oscillations and avoiding the strong overestimation produced by the Scharfetter–Gummel flux, although limiter-induced dissipation may still lead to underestimation of the electron density and ionization source term in steep-gradient regions. These findings demonstrate that spatial grid resolution and interfacial flux discretization should not be regarded as minor implementation details in RF-CCP fluid simulations. For coarse-grid simulations, multidimensional modeling, and large-scale parameter scans, the reliability of a fluid solution should be assessed by jointly examining electron flux, sheath evolution, electron power absorption, mean electron temperature, ionization source term, electron density, and current density.