The constructive approach to the numerical solution of quasi-linear advection equations

The constructive approach to the construction of the conservative difference scheme is discussed for quasi-linear advection equation. Construction of the scheme is based on the local linear spline-reconstruction of the numerical solution within the computational cell. The numerical scheme does not have any artificial regularizations, has the second order accuracy on smooth solutions and is the quasi-monotone in the vicinity of discontinuities. The procedure of generalizing the approach to the gas dynamic equations is discussed. The examples of test calculations and also the example of calculation of the actual application problem of the interaction of gas jets which are taken off from the aircraft engine with the reflective screen-bump which was carried out under the scheme are presented.