Improved weighted ENO scheme based on parameters involved in nonlinear weights

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 L1− 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.