cient Implementation of Weighted ENO Schemes 1

In this paper, we further analyze, test, modify and improve the high order WENO (weighted essentially non-oscillatory) nite diierence schemes of Liu, Osher and Chan 9]. It was shown by Liu et al. that WENO schemes constructed from the r th order (in L 1 norm) ENO schemes are (r +1) th order accurate. We propose a new way of measuring the smoothness of a numerical solution, emulating the idea of minimizing the total variation of the approximation, which results in a 5 th order WENO scheme for the case r = 3, instead of the 4 th order with the original smoothness measurement by Liu et al. This 5 th order WENO scheme is as fast as the 4 th order WENO scheme of Liu et al. and, both schemes are about twice as fast as the 4 th order ENO schemes on vector supercomputers and as fast on serial and parallel computers. For Euler systems of gas dynamics, we suggest to compute the weights from pressure and entropy instead of the characteristic values to simplify the costly characteristic procedure. The resulting WENO schemes are about twice as fast as the WENO schemes using the characteristic decompositions to compute weights, and work well for problems which do not contain strong shocks or strong reeected waves. We also prove that, for conservation laws with smooth solutions, all WENO schemes are convergent. Many numerical tests, including the 1D steady state nozzle ow problem and 2D shock entropy wave interaction problem, are presented to demonstrate the remarkable capability of the WENO schemes, especially the WENO scheme using the new smoothness measurement, in resolving complicated shock and ow structures. We have also applied Yang's artiicial compression method to the WENO schemes to sharpen contact discontinuities.