We study the cosmological Burgers model, as we call it, which is a nonlinear hyperbolic balance law (in one and two spatial variables) posed on an expanding or contracting background. We design a finite volume scheme that is fourth-order in time and second-order in space, and allows us to compute weak solutions containing shock waves. Our main contribution is the study of the asymptotic structure of the solutions as the time variable approaches infinity (in the expanding case) or zero (in the contracting case). We discover that a saddle competition is taking place which involves, on one hand, the geometrical effects of expanding or contracting nature and, on the other hand, the nonlinear interactions between shock waves.

Sustainable management of aquatic ecosystems requires effective strategies to monitor and mitigate microplastic pollution, particularly in vulnerable tidal river systems. Microplastic accumulation in these environments poses significant environmental risks, threatening biodiversity, ecosystem health, and long-term water quality. This study employs a three-dimensional hydrodynamic model (TELEMAC-3D—v8p5) coupled with a Lagrangian particle tracking model (CaMPSim-3D—v1.2.1) to simulate microplastic transport dynamics in the lower Fraser River, British Columbia, Canada. The model incorporates tidal forcing, riverine hydrodynamics, and mixing processes, and was validated with good agreement against observed water levels. This model provides a high-resolution representation of microplastic dispersion under varying release scenarios, including emissions from combined sewer overflows (CSOs) and wastewater treatment plants (WWTPs). A novel approach is proposed to identify microplastic accumulation zones using the OPTICS (Ordering Points to Identify the Clustering Structure) clustering algorithm. Accumulation zone locations remain spatially consistent despite variations in release volume. Persistent clusters occurred near channel constrictions and shoreline segments associated with flow deceleration. These findings demonstrate the robustness of the method and provide a systematic framework for prioritizing high-risk areas, supporting targeted monitoring and informing sustainable estuarine management.

A deeper understanding of the transport processes and pathways of pollutants and plastic debris in aquatic environments, such as water bodies, riverine and coastal zones, is required to effectively understand, prevent, and mitigate marine pollution. In this study, we present an updated version of the high-performance, three-dimensional Canadian Microplastics Simulation (CaMPSim-3D) particle tracking model (PTM), focused on simulating microplastics transport in marine and riverine environments. CaMPSim-3D is an efficient tool that utilizes the principles of ray tracing to enhance the accuracy and performance of particle simulations. New features of the PTM include the implementation of free-slip boundary conditions, particle settling and resuspension mechanisms, and turbulent diffusion, which improve the solution of the diffusion equations. A novel method for computing gradients over triangular prism elements and a double ray casting approach to solve the diffusion equations are also proposed. Additionally, accurate particle settling velocity for circular and elliptical microplastic fibres has been incorporated into the model. Several test cases are presented to demonstrate the new features of the PTM. A sensitivity analysis is performed to investigate the effects of different boundary conditions, particle settling velocity, and turbulent diffusion on the transport and fate of microplastics at two sites: the Saguenay Fjord and the Ottawa River. A test case for model validation against field sampling is also presented for the Ottawa River. The results show that the PTM is efficient, and the use of turbulent diffusion and settling/resuspension mechanisms is necessary, particularly when simulating sinking particles with higher density than water. The findings of this study provide valuable insights into the transport processes of microplastics in aquatic environments and can be used to improve the accuracy of microplastics simulations as well as to identify microplastics accumulation areas.