Numerical smoothness and error analysis for RKDG on the scalar nonlinear conservation laws

The new concept of numerical smoothness is applied to RKDG methods on the scalar nonlinear conservation laws. The main result is an a posteriori error estimate for the RKDG methods of arbitrary order in space and time, with optimal convergence rate. In this paper, the case of smooth solutions is the focus point. However, the error analysis framework is prepared to deal with discontinuous solutions in the future.