The Euler equations are used to describe inviscid and compressible fluids, and in doing so all the physical dissipations, mass diffusion, viscosity and heat conduction, are ignored under the consideration that they are extremely weak. In numerical simulations of inviscid and compressible fluids these missing physical dissipations are substituted by the numerical dissipations introduced by the schemes for numerical stability. These numerical dissipations are different from the real physical dissipations in both form and magnitude. The numerical dissipations will cause nonphysical phenomena in simulations, such as in simulations of flows with large density and pressure ratios and interfacial instabilities. Our solution to the problem is first to develop numerical schemes without dissipations and then to introduce numerical dissipations in the form of the missing physical dissipations to maintain the stability of the schemes. The research is ongoing.