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The main contribution of this work is to construct higher than second orderaccurate total variation diminishing (TVD) schemes which can preserve highaccuracy at non-sonic extrema with out induced local oscillations. It is donein the framework of local maximum principle (LMP) and non-conservativeformulation. The representative uniformly second order accurate schemes areconverted in to their non-conservative form using the ratio of consecutivegradient. These resulting schemes are analyzed for their non-linear LMP/TVDstability bounds using the local maximum principle. Based on the bounds, secondorder accurate hybrid numerical schemes are constructed using a shock detector.Numerical results are presented to show that such hybrid schemes yield TVDapproximation with second or higher order convergence rate for smooth solutionwith extrema.

Local principle satisfying high order total variation diminishing approximation for non-sonic data extrema