The finite difference integration methods of Rusanov, Lax-Wendroff, Rubin and Burstein, MacCormack, Hyman, the Flux-Corrected Transport (FCT)-SHASTA Phoenical, the FCT-SHASTA Phoenical Low Phase Error schemes of Boris and Book, the Hybrid scheme of Harten and Zwas, and the Artificial Compression Method of Harten are tested. These methods are utilized to solve the nonlinear hyperbolic equations describing propagation of finite amplitude waves, wave steepening, and shock formation and propagation in a closed-end tube for many wave cycles. The resulting pressure oscillation data is spectrally analyzed. For “traditional” second-order, nonmonotonic schemes, it is shown that the factors which cause the initial post-shock “wiggles” eventually lead to the generation of highly erroneous solutions. Two modified second-order schemes, FCT-SHASTA and a combination of the Lax-Wendroff, Hybrid, and Artificial Compression schemes, produced the best shock resolution and harmonic content. Finally, it is shown that the FCT schemes yield erroneous solutions at high wave amplitudes.