Using one-dimensional hydrodynamic simulations including interstellar heating, cooling, and thermal conduction, we investigate nonlinear evolution of gas flow across galactic spiral arms. We model the gas as a non-self-gravitating, unmagnetized fluid, and follow its interaction with a stellar spiral potential in a local frame comoving with the stellar pattern. Initially uniform gas rapidly separates into warm and cold phases as a result of thermal instability (TI), and also forms a quasi-steady shock that prompts phase transitions. After saturation, the flow follows a recurring cycle: warm and cold phases in the interarm region are shocked and immediately cool to become a denser cold medium in the arm; post-shock expansion reduces the mean density to the unstable regime in the transition zone and TI subsequently mediates evolution back into warm and cold interarm phases. For our standard model with n_0 = 2 cm^-3, the gas resides in the dense arm, thermally-unstable transition zone, and interarm region for 14%, 22%, 64% of the arm-to-arm crossing time. These regions occupy 1%, 16%, and 83% of the arm-to-arm distance, respectively. Gas at intermediate temperatures represents ~25-30% of the total mass, similar to the fractions estimated from HI observations. Despite transient features and multiphase structure, the time-averaged shock profiles can be matched to that of a diffusive isothermal medium with temperature 1,000 K and particle mean free path of l_0 = 100 pc. Finally, we quantify numerical conductivity associated with translational motion of phase-separated gas on the grid, and show that convergence of numerical results requires the numerical conductivity to be comparable to or smaller than the physical conductivity. (Abridged)
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