SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1071-A1096, January 2020.
In this paper, we develop the ball-approximated characteristics (B-char) method, which is an algorithm for efficiently implementing characteristic-based schemes in two and three dimensions. Core to the implementation of numerical schemes is the evaluation of integrals, which in the context of characteristic-based schemes with piecewise constant approximations boils down to computing the intersections between two regions. In the literature, these regions are approximated by polytopes (polygons in two dimensions and polyhedra in three dimensions) and, due to this, the implementation in three dimensions is nontrivial. The main novelty in this paper is the approximation of the regions by balls, whose intersections are much cheaper to compute than those of polytopes. Of course, balls cannot fully tessellate a region, and hence some mass may be lost. We perform some adjustments, and also solve an optimization problem, in order to yield a scheme that is both locally and globally mass conserving. This algorithm can achieve results that are similar to those obtained from an implementation which uses polytopal intersections, with a much cheaper computational cost.

中文翻译：

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二维和三维中基于质量守恒特征的方案的有效实现

SIAM科学计算杂志，第42卷，第2期，第A1071-A1096页，2020年1月。
在本文中，我们开发了球近似特征（B-char）方法，该方法是一种在二维和三维中有效实现基于特征的方案的算法。数值方案实施的核心是积分的评估，在具有分段常数近似值的基于特征的方案的上下文中，积分可以归结为计算两个区域之间的交点。在文献中，这些区域由多面体（二维的多边形和三维的多面体）近似，因此，在三维上的实现是不平凡的。本文的主要新颖之处在于用球近似区域，其交点的计算比多面体的交点便宜得多。当然，球无法完全细分一个区域，因此可能会丢失一些质量。我们进行一些调整，并解决一个优化问题，以产生一个在本地和全局范围内均保持质量的方案。该算法可以获得的结果与从使用多角形交叉点的实现中获得的结果相似，但计算成本却便宜得多。