A large concern in numerical methods for solving hyperbolic conservation laws is to create a scheme that is sufficiently high-order to be able to capture shock in fluid flows without giving rise to non-physical pre- and post-shock oscillations. We aim to build a new high-order interpolation scheme which maintains convergence for smooth profiles while accurately capturing sharp discontinuities in solution and does not require an unnecessarily large stencil or case-dependent limiters.

Therefore we propose a conservative multi-moment method based on 3rd order polynomial interpolation functions and the essentially non-oscillatory (ENO) scheme. Based on CIP-CSL3, the proposed method employs two multi-moment based 3rd order polynomial interpolation functions; CIP-CSL3D where three constraints are used in the upwind cell with one constraint in the downwind cell-centre, and a new, complementary method CIP-CSL3U, which uses a constraint in the upwind cell-centre instead. Based on the ENO (Essentially Non-Oscillatory) scheme, either CSL3D or CSL3U is chosen using a local smoothness indicator.

Numerical results from common benchmark tests in one dimension (both linear and non-linear) show that this scheme maintains 4th order accuracy for smooth profiles while successfully reducing numerical oscillation around discontinuity without giving way to numerical diffusion. Its competitive results show that this is a promising high-order method for CFD applications given its algorithmic simplicity, and can be extended for multi-dimensions.