Stability investigation of an implicit difference scheme for a nonlinear transport equation

In this paper, we investigate the stability with respect to initial data in the uniform norm of an implicit difference scheme approximating a nonlinear transport equation. An iterative process is used to implement the difference scheme. The convergence of the iterative process and the stability of the difference scheme are proven in the case of initial data guaranteeing the absence of shock waves. In the case of the occurrence of shock waves, estimates of the growth of spatial derivatives at each time layer are obtained. An adaptive computational algorithm for solving the transfer equation during the formation of shock waves is built.