Analytical solutions to detect the scheme dispersion for the coupled nonlinear equations

It is shown, how even particular traveling wave asymptotic solution may describe the defects on the shock wave profile caused by the dispersion features of the numerical scheme of the coupled nonlinear gas dynamics equations. For this purpose the coupled nonlinear partial differential equations or the so-called differential approximation of the scheme, are obtained, and a simplification of the method of differential approximation is suggested to obtain the desired asymptotic solution. The solution is used to study the roles of artificial viscosity and the refinement of the mesh for the suppression of the dispersion of the scheme.