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Summary The main contribution of this work is to classify the solution region including data extrema for which high‐order non‐oscillatory approximation can be achieved. It is performed in the framework of local maximum principle (LMP) and non‐conservative formulation. The representative uniformly second‐order accurate schemes are converted in to their non‐conservative form using the ratio of consecutive gradients. Using the local maximum principle, these non‐conservative schemes are analyzed for their non‐linear LMP/total variation diminishing stability bounds which classify the solution region where high‐order accuracy can be achieved. Based on the bounds, second‐order accurate hybrid numerical schemes are constructed using a shock detector. The presented numerical results show that these hybrid schemes preserve high accuracy at non‐sonic extrema without exhibiting any induced local oscillations or clipping error. Copyright © 2015 John Wiley & Sons, Ltd.

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In this article, novel smoothness indicators are presented for calculating the nonlinear weights of weighted essentially non-oscillatory scheme to approximate the viscosity numerical solutions of Hamilton-Jacobi equations. These novel smoothness indicators are constructed from the derivatives of reconstructed polynomials over each sub-stencil. The constructed smoothness indicators measure the arc-length of the reconstructed polynomials so that the new nonlinear weights could get less absolute truncation error and gives a high-resolution numerical solution. Extensive numerical tests are conducted and presented to show the performance capability and the numerical accuracy of the proposed scheme with the comparison to the classical WENO scheme.

In this article, novel smoothness indicators are presented for calculating the nonlinear weights of the weighted essentially non-oscillatory scheme to approximate the viscosity numerical solutions of Hamilton-Jacobi equations. These novel smoothness indicators are constructed from the derivatives of reconstructed polynomials over each sub-stencil. The constructed smoothness indicators measure the arc-length of the reconstructed polynomials so that the new nonlinear weights could get less absolute truncation error and give a highresolution numerical solution. Extensive numerical tests are conducted and presented to show the performance capability and the numerical accuracy of the proposed scheme with the comparison to the classical WENO scheme.