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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.

In this article an efficient sixth-order finite difference weighted essentially non-oscillatory scheme is developed to solve nonlinear degenerate parabolic equations. A new type of nonlinear weights are constructed with an introduction of a global smoothness indicator by a linear combination of local derivatives information involved in the smaller stencils that are developed with the help of generalized undivided differences in L1-norm. The advantage with these local smoothness indicators is that, each derivative involved in these are of higher order accurate as compared to the smoothness indicators developed by Liu et al. [SIAM, J. Sci. Comput, 2011]. Based on method of lines, we use strong stability preserving third-order Runge-Kutta (SSP-RK) scheme to evaluate time derivative. Various numerical tests are conducted in one and two dimensions to demonstrate the performance enhancement, resolution power and numerical accuracy of the scheme. The proposed scheme provides a better agreement in achieving the numerical solution as compared to the schemes developed by Liu et al. [SIAM, J. Sci. Comput., 2011] and Hajipour and Malek [Appl. Math. Model., 2012].