In the present paper, an exact difference scheme for the initial boundary- value problem of the third kind for an inhomogeneous hyperbolic equation of the second order with constant coefficients has been constructed on ordinary rectangular grids with constant space and time steps, where the Courant number γ=1. Later we proved a priori estimates of the stability in energy norm. For a quasi-linear wave equation on the moving characteristic grid a difference scheme has been constructed, which has the second order of approximation for the initial boundary-value problem and is exact for the Cauchy problem. The computational results for smooth functions and for a weak solution confirm the high accuracy of the introduced algorithm. We have also constructed exact difference schemes for the Cauchy problem for a system of two hyperbolic equations of the first order with constant coefficients on grids with constant space and time steps. Stability in energy norm for one of the constructed schemes has been proved. Using a method analogous to that used for the nonlinear wave equation a difference scheme for a nonlinear gas dynamic system has been constructed.