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.

A solution to the overdamping problem when simulating dust–gas mixtures with smoothed particle hydrodynamics

ABSTRACT We present a fix to the overdamping problem found by Laibe & Price when simulating strongly coupled dust–gas mixtures using two different sets of particles using smoothed particle hydrodynamics. Our solution is to compute the drag at the barycentre between gas and dust particle pairs when computing the drag force by reconstructing the velocity field, similar to the procedure in Godunov-type solvers. This fixes the overdamping problem at negligible computational cost, but with additional memory required to store velocity derivatives. We employ slope limiters to avoid spurious oscillations at shocks, finding the van Leer Monotonized Central limiter most effective.

A MUSTA-FORCE Algorithm for Solving Partial Differential Equations of Relativistic Hydrodynamics

Understanding event-by-event correlations and fluctuations is crucial for the comprehension of the dynamics of heavy ion collisions. Relativistic hydrodynamics is an elegant tool for modelling these phenomena; however, such simulations are time-consuming, and conventional CPU calculations are not suitable for event-by-event calculations. This work presents a feasibility study of a new hydrodynamic code that employs graphics processing units together with a general MUSTA-FORCE algorithm (Multi-Stage Riemann Algorithm – First-Order Centred Scheme) to deliver a high-performance yet universal tool for event-by-event hydrodynamic simulations. We also investigate the performance of selected slope limiters that reduce the amount of numeric oscillations and diffusion in the presence of strong discontinuities and shock waves. The numerical results are compared to the exact solutions to assess the code’s accuracy.

A Novel Multi-Dimensional Limiter for High-Order Finite Volume Methods on Unstructured Grids

This paper proposes a novel distance derivative weighted ENO (DDWENO) limiter based on fixed reconstruction stencil and applies it to the second- and highorder finite volume method on unstructured grids. We choose the standard deviation coefficients of the flow variables as the smooth indicators by using the k-exact reconstruction method, and obtain the limited derivatives of the flow variables by weighting all derivatives of each cell according to smoothness. Furthermore, an additional weighting coefficient related to distance is introduced to emphasize the contribution of the central cell in smooth regions. The developed limiter, combining the advantages of the slope limiters and WENO-type limiters, can achieve the similar effect of WENO schemes in the fixed stencil with high computational efficiency. The numerical cases demonstrate that the DDWENO limiter can preserve the numerical accuracy in smooth regions, and capture the shock waves clearly and steeply as well.