International Journal of Mathematical, Engineering and Management Sciences Vol. 5, No. 4, 631-642, 2020 https://doi.org/10.33889/IJMEMS.2020.5.4.051 631 Discontinuity Preserving Scheme Arun Govind Neelan Department of Aerospace Engineering, Indian Institute of Space Science and Technology, Thiruvananthapuram - 695547, Kerala, India. Corresponding author: arunneelaniist@gmail.com Manoj T. Nair Department of Aerospace Engineering, Indian Institute of Space Science and Technology, Thiruvananthapuram - 695547, Kerala, India. E-mail: manojtnair@iist.ac.in (Received December 23, 2019; Accepted April 14, 2020) Abstract Non-linear schemes are widely used in high-speed flows to capture the discontinuities present in the solution. Limiters and weighted essentially non-oscillatory schemes (WENO) are the most common non-linear numerical schemes. Most of the high-resolution schemes use the piecewise parabolic reconstruction (PPR) technique for their robustness. However, it may be impossible to achieve non-oscillatory and dissipation-free solutions with the PPR technique without non-linear switches. Most of the shock-capturing schemes use excessive dissipation to suppress the oscillations present in the discontinuities. To eliminate these issues, an algorithm is proposed that uses the shock-capturing scheme (SCS) in the first step, and then the result is refined using a novel scheme called the Discontinuity Preserving Scheme (DPS). The present scheme is a hybrid shock capture-fitting scheme. The present scheme has outperformed other schemes considered in this work, in terms of shock resolution in linear and non-linear test cases. The most significant advantage of the present scheme is that it can resolve shocks with three grid points. Keywords- Shock capturing scheme, WENO, High-resolution schemes, Conservative schemes, Finite volume method. 1. Introduction In the finite volume framework, shock-capturing scheme (SCS) is often implemented using a flux difference splitting approach. The two essential components are suitable reconstruction operator to determine approximate left and right states, and an approximate Riemann solver to obtain the numerical flux. The numerical dissipation is present in both these steps. Unlike in the case of a smooth solution, obtaining a higher-order result using a linear reconstruction is not possible in the presence of discontinuities. The order barrier theorem of Godunov (1959) stated that all schemes higher than the first-order are non-monotonous. The use of first-order schemes would result in excessive dissipation and smearing of shocks. Van Leer (1979) extended the SCS to second-order for shock problems using the concept of Total Variation Diminishing (TVD). The TVD schemes make use of nonlinear switching functions. However, they reduce to lower-order at the shocks and suffer from loss of accuracy in non-smooth extrema. Because TVD schemes introduce excessive dissipation and reduce to lower-order at shocks, they give diffused results. These limitations were partially overcome by Harten et al. (1987). By the introduction of essentially non-oscillatory (ENO) scheme. Under this scheme, the solution's smoothness is evaluated on several stencils, and then the flux is determined based on the smoothest stencil that eliminates interpolating through discontinuity. These schemes are nonlinear