Abstract: We study a new edge stabilization method for the finite element discretization of the convection-dominated diffusion-convection equations. In addition to the stabilization of the jump of the normal derivatives of the solution across the inter-element-faces, we additionally introduce a SUPG/GaLS-like stabilization term but on the domain boundary other than in the interior of the domain. New stabilization parameters are also designed. Stability and error bounds are obtained. Numerical results are presented. Theoretically and numerically, the new method is much better than other edge stabilization methods and is comparable to the SUPG method, and generally, the new method is more stable than the SUPG method.

Abstract: The behavior prediction of the transport system of a region and optimization of such a system in connection with the data networks, especially for large-scale areas, seems very necessary and quite challenging problem and attracts an attention of many researchers [1]. Due to extensive expansion of cities, increase of the number of vehicles on the roads and constant grow of the transportation lines with increased complexity, this task is up-to-date and hard to solve with the methods presented at the moment. Empirical correlations or data obtained during observations of some crucial areas of interest are not always enough to predict parameters of that networks at different states, critical situations or moments of time, therefore numerical simulation seems to be a good choice to predict traffic flow behavior. It should be mentioned that numerical methods used earlier are not always capable to deal with the large-scale problems due to dependence on the extensive computational resources or inability to resolve local features of the transport flow and data networks [2]. Existing mathematical models can be divided into two classes: a) local microscopic models which are working on car-by-car basis [3] and b) continuum-like models which are dealing with the entire traffic flow and operating with the averaged quantities [4]. The first class is capable to resolve interaction between different vehicles, data transfer between cars or between a car and a road infrastructure, thus providing detailed information in critical areas (crossroads, joints, etc.), but requires significant amount of computational resources as simulated domain grows. The continuum approach, on the other hand, is much less resource demanding and suitable to simulate large-scale road elements (long parts of the roads, for example), but unable to resolve complex parts of the transportation system without some additional knowledge about traffic flow in that regions obtained “a-priori”. Presented work is focused on the overcoming of two deficiencies of both classes by the proposal of a hybrid modeling approach. This approach utilizes both types of methods at different scales. For the local (small) scale the microscopic models are used to obtain distribution of traffic parameters at particular road elements and provide network data distribution between road users. The integration procedure is performed for car parameters such as density and velocity and averaged values is substituted into the continuum-like model based on the hydrodynamic approach. That part of the algorithm performs the simulation of the entire transport network at macro scales without detailed description of local elements, and output values obtained at that step are used as input values for the microscopic model. Therefore, an iterative technique is performed to obtain developed transport flow for the entire region. In parallel, the traffic flow information will be used to predict the load of the data networks which are used for communication purposes between vehicles and road infrastructure and evaluation of data payload optimization will be conducted to provide sufficient throughput and reliability of such networks. The algorithm is implemented on the basis of high level programming language and initial simulations for the purpose of the testing and validation will be conducted. These simulations will be compared along with the experimental data obtained by the means of the experimental apparatus initially prepared by the research group from SPbSPU. The paper is presented within the framework of the project No. 18-07-00430 which is supported by the Russian Foundation for Basic Research. 1. Xinkai Wu, Henry X. Liu, Using high-resolution event-based data for traffic modeling and control: An overview, Transportation Research Part C, 2014, Vol. 42, P. 28–43. 2. M. Kontorinaki, A. Spiliopoulou, C. Roncoli, M. Papageorgiou, First-order traffic flow models incorporating capacity drop: Overview and real-data validation, Transportation Research Part B, 2017, Vol. 106, P. 52-75. 3. Nagel K., Schreckenberg M. A cellular automaton model for freeway traffic. Phys. I France, 1992, Vol. 2, P. 2221–9. 4. C. Wagner, A Navier-Stokes-like traffic model, Physica A, 1997, Vol. 245, P. 124-38.

Abstract: An conservative remapping scheme often requires intersections between two mesh and a reconstruction scheme on the old cells (Lagrangian mesh). Computing the exact overlaps is complicated even in the simplest case. In this paper, We propose method to calculate intersections of two dismissable general quadrilateral mesh of the same logically structure in a planar domain. The quadrilateral polygon intersection problem is reduced to a problem that how an edge in a new mesh intersects with a local frame which consists at most 7 connected edges in the old mesh. As such, locality of the method is persevered. The alternative direction technique is applied to reduce the dimension of the searching space, We call the method as a symmetric sweep algorithm. It reduces more than 256 possible intersections between a new cell with the old mesh to 34 (17 when considering symmetry) programmable intersections between an edge and an local frame whenever the intersection between the old and new cell does not degenerate. Besides, we shall show how the computational amount depends on the underlying problem in term of singular intersection points. A simple and detailed classification on the type of overlaps is presented. According to classification, degeneracy of an intersection can be easily identified.

Abstract: This paper presents a finite element solver for linear poroelasticity problems on quadrilateral meshes based on the displacement-pressure two-field model. This new solver combines the Bernardi-Raugel element for linear elasticity and a weak Galerkin element for Darcy flow through the implicit Euler temporal discretization. The solver does not use any penalty factor and has less degrees of freedom compared to other existing methods. The solver is free of nonphysical pressure oscillations, as demonstrated by numerical experiments on two widely tested benchmarks. Extension to other types of meshes in 2-dim and 3-dim is also discussed.

Abstract: In this paper, we improve the preprocessing phase of the ALT algorithm through parallelization. ALT is a preprocessing-based, goal-directed speed-up technique that uses A* (A star), Landmarks and Triangle inequality which allows fast computations of shortest paths (SP) in large-scale networks. Although faster techniques such as arc-flags, SHARC, Contraction Hierarchies and Highway Hierarchies already exist, ALT is usually combined with these faster algorithms to take advantage of its goal-directed search to further reduce the SP search calculation time and its search space. However, ALT relies on landmarks and optimally choosing these landmarks is NP-hard, hence, no effective solution exists. Since landmark selection relies on constructive heuristics and the current SP search speed-up is inversely proportional to landmark generation time, we propose a parallelization technique which cuts the landmark generation time significantly while increasing its effectiveness.