A modified monotonicity-preserving high-order scheme with application to computation of multi-phase flows

Abstract

The monotonicity preserving (MP) polynomial-based reconstruction scheme with Runge-Kutta time stepping by Suresh and Huynh (1997) has been applied extensively in computational fluid dynamics for solving hyperbolic partial differential equations and other convection-dominated problems. However, despite of its algorithmic simplicity in achieving a desired high-order of accuracy, the MP scheme is shown to lose the accuracy and robustness for a long-term computation of flows containing complex smooth structures interspersed with discontinuities. In this paper, the original MP scheme is first reviewed with a focus on the potential causes that lead to its failure. We then propose an improved MP scheme that can overcome the weakness in the MP scheme. The proposed MP scheme is constructed so that it can bypass smooth extrema without loss of accuracy while preserving monotonicity and properly reducing the order of accuracy near discontinuities. Various challenging scalar advection test cases are considered to demonstrate the performance of the presented method. Numerical results reveal that it achieves the highly accurate solutions in smooth parts of the flow and yields a significant improvement in accuracy and shape-preserving property. Moreover, the successful application to several classes of two- and three-dimensional unsteady multi-phase flows highlights the potential of the present method for resolving both sharp and complex interface deformations.