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The advantage of WENO-JS5 scheme [ J. Comput. Phys. 1996] over the WENO-LOC scheme [J. Comput. Phys.1994] is that the WENO-LOC nonlinear weights do not achieve the desired order of convergence in smooth monotone regions and at critical points. In this article, this drawback is achieved with the WENO-LOC smoothness indicators by constructing a WENO-Z type nonlinear weights which contains a novel global smoothness indicator. This novel smoothness indicator measures the derivatives of the reconstructed flux in a global stencil, as a result, the proposed numerical scheme could decrease the dissipation near the discontinuous regions. The theoretical and numerical experiments to achieve the required order of convergence in smooth monotone regions, at critical points, the essentially non-oscillatory (ENO), the analysis of parameters involved in the nonlinear weights like $\epsilon$ and $p$ are studied. From this study, we conclude that the imposition of certain conditions on $\epsilon$ and $p$, the proposed scheme achieves the global order of accuracy in the presence of an arbitrary number of critical points. Numerical tests for scalar, one and two-dimensional system of Euler equations are presented to show the effective performance of the proposed numerical scheme., Comment: 25 pages, 10 figures

Here, we have analyzed the weights of the fifth-order finite difference weighted essentially non-oscillatory WENO-P scheme developed by Kim et al. (J. Sci. Comput. 2016) to approximate the solutions of hyperbolic conservation laws. The main ingredient of WENO schemes is the construction of smoothness indicators, which resolves odd behavior of the scheme near discontinuities. In WENO-P, the smoothness indicators are constructed in L 1 − norm. It is observed that analytically as well as numerically, the WENO-P weights do not achieve required ENO order of accuracy near discontinuities. To recover the desired order of accuracy, we have imposed some constraints on the weight parameters to guarantee that the WENO-P scheme achieves the desired ENO order of accuracy near discontinuities and have the over all fifth-order accuracy in smooth regions of solutions with an arbitrary number of vanishing derivatives. Numerical results are presented with the new weights to verify the robustness and accuracy of the proposed scheme for one and two-dimensional system of Euler equations.

This paper deals with a new fifth-order weighted essentially non-oscillatory (WENO) scheme improving the WENO-NS and WENO-P methods which are introduced in Ha et al. J. Comput. Phys. (2013) and Kim et al., J. Sci. Comput. (2016) respectively. These two schemes provide the fifth-order accuracy at the critical points where the first derivatives vanish but the second derivatives are non-zero. In this paper, we have presented a scheme by defining a new global-smoothness indicator which shows an improved behavior over the solution to the WENO-NS and WENO-P schemes and the proposed scheme attains optimal approximation order, even at the critical points where the first and second derivatives vanish but the third derivatives are non-zero., 23 pages, 14 figures