Good Neighborhoods for Multidimensional Van Leer Limiting

• Published in: Journal of computational physics 1999-09, Vol.154 (1), p.237-241
• Main Author: Swartz, Blair
• Format: Article
• Language: English
• Subjects: FUNCTIONS; LIMITING VALUES; MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; MESH GENERATION; NUMERICAL SOLUTION
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1006/jcph.1999.6308

Van Leer limiting uses nearby cell-means of a function (integral mean-values--weighted by a prescribed positive density--that are taken over each of a collection of nearby computational cells) to restrict the range of values allowed to a linear approximation of the function on a given central cell. These nearby cells--whose cell-means are actually involved in the limiting--are called the central cell's neighbors; and the set of these neighbors is called the central cell's neighborhood. The use of certain neighborhoods in multidimensional Van Leer limiting can force even linear functions to be subject to restriction over the central cell. A simple geometric property characterizes those neighborhoods whose use would not require that any linear functions be limited. (Such a neighborhood is called a good neighborhood for Van Leer limiting since its use would not preclude second-order accuracy in the local linear approximation of a smooth function by one that is Van Leer limited--unless the additional, here unspecified, details for obtaining the approximation preclude it by themselves.) The characterization is as follows, where it is presumed that the cells lie in a finite dimensional vector space: One has chosen a good neighborhood for a given central cell if and only if the convex hull of the centroids of its associated neighbors contains that central cell. Details are given.