This dissertation proposes improvements in the numerical algorithm for the dynamics of flows that involve discontinuities and broadband fluctuations simultaneously. These two flow features suggest numerical strategies of a paradoxical nature because the discontinuities demand large dissipation and the small-scale smooth features require the opposite. There may be several ways to approach such a complicated issue, such as combining a shock-capturing scheme with a low dissipative method using a shock detector, but the natural choice, to avoid the redundancy of using the shock detector and the stability issue as a result of coupling, is a numerical technique that can adjust adaptively with flow regimes. The weighted essentially non-oscillatory (WENO) scheme may be this choice. However, there are two sources of dissipation associated with the WENO procedure: upwind optimal stencil and nonlinear adaption mechanism. The current work suggests a robust and comprehensive treatment for the minimization of dissipation error from these two sources. The optimization technique is used to delay the dissipation of the upwind optimal stencil to those wavenumbers for which the dispersion error is large. However, optimization decreases the formal order of accuracy of the optimal stencil from fifth order to third order. Using the WENO procedure, the third-order accuracy is verified in the smooth region, except the critical point of order two, where the order of accuracy reduces to at least second order. The lost accuracy at the second-order critical point is restored in an attempt to reduce the dissipation induced by the nonlinear adaptive weights. The modification of the nonlinear weights to reduce the dissipation is introduced by redefining them with an additional smoothness indicator. Other suggestions regarding the issues to minimize the dissipation of the nonlinear weights are also reviewed.