INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids (2015) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/fld.4202 Local maximum principle satisfying high-order non-oscillatory schemes  Ritesh Kumar Dubey 1, * ,† , Biswarup Biswas 1 and Vikas Gupta 2 1 Research Institute, SRM University, Tamilnadu, India 2 LNMIIT, Jaipur, Rajasthan, India 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. Received 22 July 2015; Revised 29 September 2015; Accepted 7 November 2015 KEY WORDS: hyperbolic conservation laws; smoothness parameter; non-sonic critical point; total variation stability; finite difference schemes 1. INTRODUCTION We consider the 1D scalar conservation law associated with the conserved variable u.x;t/, @ @t u.x;t/ C @ @x f .u.x; t // D 0; .x; t/ 2 R  R C u.x; 0/ D u 0 .x/ (1) where f .u/ is a nonlinear flux function. The numerical approximation for the solution of (1) is carried out by the discretization of the spatial and temporal spaces into N equispaced cells I i D Œx i  1 2 ;x i C 1 2 ; i D 0; 1; : : : N  1 of length x and M equispaced intervals  t n ;t nC1  ; n D 0; 1; : : : ; M  1 of length t , respectively. Let x i D ix and t n D nt denote the cell center of cell I i and the n th time level, respectively, then a conservative numerical approximation for (1) can be defined by u nC1 i D u n i    F n i C 1 2  F n i  1 2  ;  D t x : (2) where u n i D u.x i ;t n / and F n i ˙ 1 2 is the numerical flux function defined at the cell interface x i ˙ 1 2 at time level n. The characteristics speed a.u/ D @f .u/ @u associated with (1) can be approximated as *Correspondence to: Ritesh Kumar Dubey, Research Institute, SRM University, Tamilnadu, India. † E-mail: riteshkd@gmail.com ‡ The carried work is financially supported by SERB-DST, India project SR/FTP/MS-015/2011. Copyright © 2015 John Wiley & Sons, Ltd.