Slope limiters have been widely used to eliminate nonphysical oscillations near discontinuities generated by finite volume methods for hyperbolic systems of conservation laws. In this study, we investigate the performance of these limiters as applied to three-dimensional modified method of characteristics on unstructured tetrahedral meshes. The focus is on the construction of monotonicity-preserving modified method of characteristics for three-dimensional transport problems with discontinuities and steep gradients in their solutions. The proposed method is based on combining the modified method of characteristics with a finite element discretization of the convection equations using unstructured grids. Slope limiters are incorporated in the method to reconstruct a monotone and essentially nonoscillatory solver for three-dimensional problems at minor additional cost. The main idea consists in combining linear and quadratic interpolation procedures using nodes of the element where departure points are localized. We examine the performance of the proposed method for a class of three-dimensional transport equations with known analytical solutions. We also present numerical results for a transport problem in three-dimensional pipeline flows. In considered test problems, the proposed method demonstrates its ability to accurately capture the three-dimensional transport features without nonphysical oscillations.

中文翻译：

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非结构化四面体网格特征的三维保持单调性修正方法

斜率限制器已被广泛用于消除由守恒定律的双曲线系统的有限体积方法产生的不连续性附近的非物理振荡。在这项研究中，我们研究了这些限制器在应用于非结构化四面体网格特征的三维修正方法时的性能。重点是构建具有不连续性和陡峭梯度的三维输运问题的单调性保持修正的特征方法。所提出的方法基于将改进的特性方法与使用非结构化网格的对流方程的有限元离散化相结合。该方法中包含了斜率限制器，可以以较小的额外成本为三维问题重建单调且本质上非振荡的求解器。主要思想在于使用元素的节点来组合线性和二次插值过程，其中出发点被定位。我们检查了所提出的方法在具有已知解析解的一类三维传输方程中的性能。我们还提供了三维管道流中传输问题的数值结果。在考虑的测试问题中，所提出的方法证明了它能够在没有非物理振荡的情况下准确捕获三维传输特征。主要思想在于使用元素的节点来组合线性和二次插值过程，其中出发点被定位。我们检查了所提出的方法在具有已知解析解的一类三维传输方程中的性能。我们还提供了三维管道流中传输问题的数值结果。在考虑的测试问题中，所提出的方法证明了它能够在没有非物理振荡的情况下准确捕获三维传输特征。主要思想在于使用元素的节点来组合线性和二次插值过程，其中出发点被定位。我们检查了所提出的方法在具有已知解析解的一类三维传输方程中的性能。我们还提供了三维管道流中传输问题的数值结果。在考虑的测试问题中，所提出的方法证明了它能够在没有非物理振荡的情况下准确捕获三维传输特征。