Author(s): GUANGSHAN JIANG, CHIWANG SHU
In this paper, we further analyze, test, modify, and improve the side. Here u 5 (u1 , ..., um), f 5 (f1 , ..., fd), x 5 (x1 , ..., xd) high order WENO (weighted essentially non-oscillatory) finite differ- and t . 0. ence schemes of Liu, Osher, and Chan. It was shown by Liu et al. WENO schemes are based on ENO (essentially non- that WENO schemes constructed from the rth order (in L1 norm) oscillatory) schemes, which were first introduced by ENO schemes are (r 1 1)th order accurate. We propose a new way of measuring the smoothness of a numerical solution, emulating Harten, Osher, Engquist, and Chakravarthy  in the form the idea of minimizing the total variation of the approximation, of cell averages. The key idea of ENO schemes is to use which results in a fifth-order WENO scheme for the case r 5 3, the ‘‘smoothest’’ stencil among several candidates to ap- instead of the fourth-order with the original smoothness measure- proximate the fluxes at cell boundaries to a high order ment by Liu et al. This fifth-order WENO scheme is as fast as the accuracy and at the same time to avoid spurious oscillations fourth-order WENO scheme of Liu et al. and both schemes are about twice as fast as the fourth-order ENO schemes on vector near shocks. The cell-averaged version of ENO schemes supercomputers and as fast on serial and parallel computers. For involves a procedure of reconstructing point values from Euler systems of gas dynamics, we suggest computing the weights cell averages and could become complicated and costly for from pressure and entropy instead of the characteristic values to multi-dimensional problems. Later, Shu and Osher [14, 15] simplify the costly characteristic procedure. The resulting WENO developed the flux version of ENO schemes which do not schemes are about twice as fast as the WENO schemes using the characteristic decompositions to compute weights and work well require such a reconstruction procedure. We will formulate for problems which do not contain strong shocks or strong reflected the WENO schemes based on this flux version of ENO waves. We also prove that, for conservation laws with smooth solu- schemes. The WENO schemes of Liu et al.  are based tions, all WENO schemes are convergent. Many numerical tests, on the cell-averaged version of ENO schemes. including the 1D steady state nozzle flow problem and 2D shock entropy wave interaction problem, are presented to demonstrate For applications involving shocks, second-order schemes the remarkable capability of the WENO schemes, especially the are usually adequate if only relatively simple structures WENO scheme using the new smoothness measurement in resolv- are present in the smooth part of the solution (e.g., the ing complicated shock and flow structures. We have also applied shock tube problem). However, if a problem contains rich Yang’s artificial compression method to the WENO schemes to sharpen contact discontinuities.