The feasibility of using optimized high-compact (OHOC) schemes for the numerical computations of nonlinear wave propagation is investigated. The high-order compact finite difference schemes which were optimized for high-resolution characteristics are less dissipative and dispersive, and they require less grid points to resolve a wave profile than the other low-order standard schemes. They are well adapted to linear problems with smooth wave solutions, however inevitably generate spurious spatial oscillations when applied to nonlinear problems with highly discontinuous wave solutions. Thus, they require an effentive artificial dissipation algorithm to damp out only the spurious oscillations, while keeping the wave components in low wavenumber range unaffected. They also require an efficient high-order low dissipation and dispersion time advancing method to produce long-time accurate nonlinear wave speeds and profiles. In this paper, for the nonlinear computations, the OHOC schemes are coupled with the artificial selective damping (ASD) terms and the fourth-order low dissipation and dispersion Runge-Kutta (LDDRK) scheme which is more efficient whthin a certain accuracy limit than the classical fourth-order Runge-Kutta scheme. It is shown that the application of these schemes to the numerical computations of the nonlinear wave propatation presents time accurate solutions with well-resolved shocks or contact surfaces and without spurious oscillations.