Multi-dimensional limiting process for finite volume methods on unstructured grids

Abstract

This paper deals with a robust, accurate and efficient multi-dimensional limiting strategy on three-dimensional unstructured grids within the framework of finite volume method. The present limiting strategy is on the line of continuous efforts to extend the multi-dimensional limiting process (MLP) onto three-dimensional tetrahedral grids, which was originally proposed on structured and triangular grids. In previous works, it was observed that the MLP limiting shows several superior characteristics, such as efficient control of multi-dimensional oscillations and accurate capture of both discontinuous and continuous multi-dimensional flow features, on triangular as well as structured grids. The design principle of the MLP limiters is based on the multi-dimensional limiting condition and the maximum principle, which can ensure multi-dimensional monotonicity through the global/local L∞ stability. Consequently, it can be shown that the MLP limiting does satisfy the local extremum diminishing (LED) condition in a truly multi-dimensional way. The present MLP slope limiters are formulated into the setting of the three-dimensional Euler system, and are refined to improve convergence characteristics for steady state problems without compromising the accuracy of computed results. Through various numerical analyses and computations, it is demonstrated that the proposed MLP limiters provide the same level of successful performances previously observed on triangular and structured grids.