or perhaps with a forcing term g(u, x, t) on the right-hand side. Here u ϭ (u 1 , ..., u m ), f ϭ (f 1 , ..., f d ), x ϭ (x 1 , ..., x d )In this paper, we further analyze, test, modify, and improve the high order WENO (weighted essentially non-oscillatory) finite differand t Ͼ 0. 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 soluschemes. The WENO schemes of Liu et al. [9] 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 For applications involving shocks, second-order schemes entropy wave interaction problem, are presented to demonstrate 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 structures as well as shocks (e.g., the shock entropy wave sharpen contact discontinuities. ᮊ 1996 Academic Press, Inc.interaction problem in Example 4, Section 8.3), high order shock capturing schemes (order of at least three) are more

Abstract. In this paper we further explore a class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in a paper by Shu and Osher, suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD (total variation diminishing) spatial discretization, verifying the claim that TVD Runge-Kutta methods are important for such applications. We then explore the issue of optimal TVD Runge-Kutta methods for second, third and fourth order, and for low storage Runge-Kutta methods.