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

In this paper, we present a weighted ENO (essentially nonoscillatory) scheme to approximate the viscosity solution of the Hamilton-Jacobi equation:This weighted ENO scheme is constructed upon and has the same stencil nodes as the third order ENO scheme but can be as high as fifth order accurate in the smooth part of the solution. In addition to the accuracy improvement, numerical comparisons between the two schemes also demonstrate that the weighted ENO scheme is more robust than the ENO scheme.

We construct, analyze, and implement a new nonoscillatory high-resolution scheme for two-dimensional hyperbolic conservation laws. The scheme is a predictor-corrector method which consists of two steps: starting with given cell averages, we first predict pointvalues which are based on nonoscillatory piecewise-linear reconstructions from the given cell averages; at the second corrector step, we use staggered averaging, together with the predicted midvalues, to realize the evolution of these averages. This results in a second-order, nonoscillatory central scheme, a natural extension of the one-dimensional second-order central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408-448]. As in the one-dimensional case, the main feature of our two-dimensional scheme is simplicity. In particular, this central scheme does not require the intricate and time-consuming (approximate) Riemann solvers which are essential for the high-resolution upwind schemes; in fact, even the computation of the exact Jacobians can be avoided. Moreover, the central scheme is "genuinely multidimensional" in the sense that it does not necessitate dimensional splitting. We prove that the scheme satisfies the scalar maximum principle, and in the more general context of systems, our proof indicates that the scheme is positive (in the sense of Lax and Liu [CFD Journal, 5 (1996), pp. 1-24]). We demonstrate the application of our central scheme to several prototype twodimensional Euler problems. Our numerical experiments include the resolution of shocks oblique to the computational grid; they show how our central scheme solves with high resolution the intricate wave interactions in the so-called double Mach reflection problem [J. Comput. Phys., 54 (1988), pp. 115-173] without following the characteristics; and finally we report on the accurate ray solutions of a weakly hyperbolic system [J. Comput. Appl. Math., 74 (1996), pp. 175-192], rays which otherwise are missed by the dimensional splitting approach. Thus, a considerable amount of simplicity and robustness is gained while achieving stability and high resolution.

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We present a general procedure to convert schemes which are based on staggered spatial grids into nonstaggered schemes. This procedure is then used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recently introduced in [H. Nessyahu and E.

Abstract. We prove a cell entropy inequality for a class of high-order discontinuous Galerkin finite element methods approximating conservation laws, which implies convergence for the one-dimensional scalar convex case.

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