Monotonicity of Multi-dimensional Limiting Process on Unstructured Grids

The present paper deals with the continuous work of extending multi-dimensional limiting process (MLP), which has been quite successfully proposed on twoand three-dimensional structured grids, onto the unstructured grids. The basic idea of the present limiting strategy is to control the distribution of both cell-centered and cell-vertex physical properties to mimic a multi-dimensional nature of flow physics, which can be formulated as so called the MLP condition. This condition satisfies the maximum principle, which ensures monotonicity, and numerical results show that MLP is effective to prevent unwanted oscillations as well as to capture multi-dimensional flow features accurately. INTRODUCTION High resolution scheme is one of the challengeable issues in hyperbolic conservation laws. Especially, discontinuities in the solution may lead spurious oscillations, which break down the numerical solution. For this reason, there have been many remarkable progresses on oscillation-free scheme, such as TVD or ENO, but most of them rely on the mathematical analysis of one-dimensional convection equation. Though this approach may work successfully in many cases, it is often insufficient or almost impossible to control oscillations near shock discontinuity in multi-dimensional flow. In order to find out the suitable criterion to prevent such oscillations in multiple dimensions, the one-dimensional monotonic condition was extended to multi-dimensional flow situations and the multi-dimensional limiting process (MLP) was successfully developed. From the series of researches, it has been clearly demonstrated that the MLP limiting strategy possesses favorable characteristics, such as enhanced accuracy and convergence behavior in numerous inviscid and viscous computations on structured grids [1, 2]. Furthermore, the MLP limiting strategy can be readily extended on unstructured grids with some modifications [3]. In this work, we explore the monotonicity of MLP on unstructured grids. After introducing the basic concept and the implementation, it is shown that the proposed scheme satisfies the maximum principle. Various numerical tests are presented to verify the performance of the proposed method. BASIC CONCEPT AND IMPLEMENTATION In order to maintain multi-dimensional monotonicity, the present limiting strategy exploits the MLP condition, which is an extension of the one-dimensional monotonic condition. On the structured grids, the MLP condition restricts the physical properties on vertex as well as cellcenter points. On vertex, a physical property is estimated by summing the monotonic variation of each coordinate direction, and then the vertex values are required to satisfy the