Numerical Smoothness and Error Analysis on Weno for the Nonlinear Conservation Laws

In this study we give an a posteriori error analysis on the WENO schemes for the nonlinear scalar conservation laws. This analysis is based on the new concept of numerical smoothness, with some new error analysis mechanisms developed for the finite difference and finite volume discretizations. The local error estimate is of optimal order in space and time. The global error estimate grows linearly in time, because of the direct application of the L1contraction between entropy solutions in the error propagation analysis. As a beginning, we only deal with smooth solutions in this paper. Within the same error propagation framework, when we deal with piecewise smooth solutions later, we only need to work on estimating the local error where smoothness is lost. The smoothness indicators not only serve the purpose of local error estimation, but also serve as a monitor on both the possible numerical instability and the expected solution shapening.