A Three-Dimensional Monotonicity-Preserving Modified Method of Characteristics on Unstructured Tetrahedral Meshes

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.