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A kinetically-based system of equations for three-phase porous media ow simulation is considered. A simple case with the following assumptions is discussed: phase transitions are absent, phases do not dissolve and do not mix, the rock compressibility is negligible. Such systems are under consideration in applied problems when the pressure changes slightly and thermal processes are absent, for example, in environmental problems. The continuity equation is modi ed via introduction of the regularizing term and the second-order time derivative. Due to conversion to the hyperbolic type the corresponding di erence equation stability is improved. An explicit algorithm is developed and adapted to high-performance computing systems. High parallelization e ciency is achieved on a classical cluster as well as on a hybrid cluster with graphics accelerators.Keywords: Multiphase ow in a porous medium, quasigasdynamic system of equations, explicit nitedi erence schemes, parallel implementation.Mathematical modelling of multiphase uid ows in porous media is necessary for solving many practically important problems, for example, for the simulation of processes of contaminant in ltration into the soil [1, 9, 10, 17]. It is well known that numerical simulation of these processes is very time-consuming and impossible without the employment of high-performance computing systems. Nowadays the rapid growth in the computer performance is mainly achieved due to the use of hybrid architectures including multicore CPUs and di erent accelerators like graphics processing units (GPU). Use of GPU for general-purpose computations is a perspective modern trend to solve large scale applied problems with high accuracy for the reasonable time. However, such architectures cause serious di culties in the software development. For example, wellknown IMPES-method for modelling porous media ows [8] assumes the solution of elliptic equation. But the Laplace operator inversion requires high computational costs and leads to decrease of parallelization efciency. Computational algorithms with logically simple structure like explicit nite-di erence schemes can be adapted easily to hybrid supercomputers and allow to exploit them more e ciently [5]. Such algorithms show perfect scaling on parallel architectures therefore they are widely used in numerical investigations of subsurface ows [7, 12].In the present paper a new algorithm for porous medium ow simulation is discussed. Currently kinetic approaches such as Lattice Boltzmann schemes [2, 7, 11] and kinetically-consistent nite di erence (KCFD) [3] schemes seem to be the most promising methods in computational uid dynamics. The new model is constructed by the analogy with KCFD schemes and the related quasigasdynamic (QGD) system of equations [13].

The paper deals with mathematical modeling of traffic flows on urban road networks. The original model is based on the cellular automata theory and presents a generalization of Nagel-Schreckenberg model to a multilane case.Numerical realization of the model is represented in a form of the program package that consists of two modules: User Interface and Visualization module (for setting initial conditions and modelling parameters and visual representation of calculations) and Computation module (for calculations).Computations are carried out for each element of the road (i.e. T or X type intersection, straight road fragment) separately and in parallel, that allows performing calculations on various complex road networks. Different kinds of average characteristics (e.g. the capacity of the crossroad) can be also obtained using the program package.

The research deals with the creation of mathematical tools for the simulation of vehicular traffic flows on complex urban transport networks using modern supercomputers. The goal of the present paper is further development of micro-and macroscopic models created by the authors earlier. The proposed 2D microscopic model is based on the cellular automata theory. In this work algorithms taking into account various driving strategies have been incorporated into the model. The model is implemented as a program package that includes User interface and Visualization module. The macroscopic model uses the continuous medium approximation: it is constructed by analogy with the quasigasdynamic system of equations. The one dimensional version is proposed in the paper, nevertheless, it allows reproducing changes in the number of lanes as well as possible road entrances and exits. Parallel algorithms adapted to high-performance computing systems have been created for both models, ensuring rapid computations on city road networks.

Abstract. The paper deals with the mathematical modeling of traffic flows on urban road networks using microscopic approach. The model is based on the cellular automata theory and presents a generalization of the Nagel-Schreckenberg model to a multilane case. The created program package allows to simulate traffic on various types of road fragments (T or X type intersection, strait road elements, etc.) and on road networks that consist of these elements. Besides that, it allows to predict the consequences of various decisions regarding road infrastructure changes, such as: number of lanes increasing/decreasing, putting new traffic lights into operation, building new roads, entrances/exits, road junctions.