  The main contribution of this work is to construct higher than second order
accurate total variation diminishing (TVD) schemes which can preserve high
accuracy at non-sonic extrema with out induced local oscillations. It is done
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
gradient. These resulting schemes are analyzed for their non-linear LMP/TVD
stability bounds using the local maximum principle. Based on the bounds, second
order accurate hybrid numerical schemes are constructed using a shock detector.
Numerical results are presented to show that such hybrid schemes yield TVD
approximation with second or higher order convergence rate for smooth solution
with extrema.
