Solving Grid Equations Using the Alternating-triangular Method on a Graphics Accelerator

The paper describes a parallel-pipeline implementation of solving grid equations using the modified alternating-triangular iterative method (MATM), obtained by numerically solving the equations of mathematical physics. The greatest computational costs at using this method are on the stages of solving a system of linear algebraic equations (SLAE) with lower triangular and upper non-triangular matrices. An algorithm for solving the SLAE with a lower triangular matrix on a graphics accelerator using NVIDIA CUDA technology is presented. To implement the parallel-pipeline method, a three-dimensional decomposition of the computational domain was used. It is divided into blocks along the y coordinate, the number of which corresponds to the number of GPU streaming multiprocessors involved in the calculations. In turn, the blocks are divided into fragments according to two spatial coordinates — x and z. The presented graph model describes the relationship between adjacent fragments of the computational grid and the pipeline calculation process. Based on the results of computational experiments, a regression model was obtained that describes the dependence of the time for calculation one MATM step on the GPU, the acceleration and efficiency for SLAE solution with a lower triangular matrix by the parallel-pipeline method on the GPU were calculated using the different number of streaming multiprocessors.The paper describes a parallel-pipeline implementation of solving grid equations using the modified alternating-triangular iterative method (MATM), obtained by numerically solving the equations of mathematical physics. The greatest computational costs at using this method are on the stages of solving a system of linear algebraic equations (SLAE) with lower triangular and upper non-triangular matrices. An algorithm for solving the SLAE with a lower triangular matrix on a graphics accelerator using NVIDIA CUDA technology is presented. To implement the parallel-pipeline method, a three-dimensional decomposition of the computational domain was used. It is divided into blocks along the y coordinate, the number of which corresponds to the number of GPU streaming multiprocessors involved in the calculations. In turn, the blocks are divided into fragments according to two spatial coordinates — x and z. The presented graph model describes the relationship between adjacent fragments of the computational grid and the pipeline calculation process. Based on the results of computational experiments, a regression model was obtained that describes the dependence of the time for calculation one MATM step on the GPU, the acceleration and efficiency for SLAE solution with a lower triangular matrix by the parallel-pipeline method on the GPU were calculated using the different number of streaming multiprocessors

Study of access template to graphics engine GM effect on the performance

The work objective is to study the effect of the graphical processor unit computational cores load level and memory access pattern on the memory bus bandwidth and scaling acceleration. The research subject is the problem of scalability of the parallel computing performance and acceleration. The following hypothesis is checked: while processing images for multi-core shared-memory systems, Gustafson - Barsis’s law is more crucial than the memory access template at the underloading of the GPU cores. The research methodology is a computational experiment with further analysis of the obtained results. The conclusions are as follows. The suggested hypothesis is proved. For that, a series of experiments on various heterogeneous computational systems with OpenCL standard support is conducted. The application field of the results obtained includes the development of algorithms and software for the highly parallel computer systems. The memory access template starts to place certain restrictions on the algorithm efficiency only when the load level of the computational cores is sufficient. Video cards with the private memory show more stable results in comparison to those which share memory with the central processing unit

Sufficient conditions for convergence of positive solutions to linearized two-dimensional sediment transport problem

Introduction. The sediment transportation is one of the major processes that define the magnitude and back surface changing rate for water bodies. The most used prognostic studies in this field are based on the mathematical models that allow reducing, and in some cases, eliminating the expensive and often environmentally burdensome experiments. Spatially one-dimensional models are mainly used to predict changes in bottom topography. For actual coastal systems with irregular coastal configuration, the flow vector is generally not orthogonal to the tangent line for the coastline at each of its points. It also may not coincide with the wind stress vector. Therefore, to solve lots of practically important problems associated with the prediction of the dynamics of the back surface of water basins, it is necessary to use spatially two-dimensional models of sediment transportation and effective numerical methods of their implementation. Materials and Methods. The authors (A.I. Sukhinov, A.E. Chistyakov, E.A. Protsenko, and V.V. Sidoryakina) have earlier proposed a spatially two-dimensional model of sediment transport that satisfies the basic conservation laws (of material balance and momentum) which is a quasilinear equation of parabolic type. The linear difference schemes are constructed and studied; model and some practically important tasks are solved. However, the theoretical study on the proximity of solutions for the original nonlinear initial-boundary value problem and the linearized continuous task - on which basis a discrete model (difference scheme) was built - remained in the shadow. The study of the linearized problem correctness and the determination of sufficient conditions for positivity of solutions are of special interest because only positive solutions to this sediment transport problem have physical value within the framework of the considered models. Research Results. The investigated nonlinear two-dimensional model of sediment transport in the coastal zone of shallow reservoirs takes account of the following physically significant factors and parameters: soil porosity; critical value of the tangent stress at which load transport starts; turbulent interaction; dynamically varying of the bottom geometry; wind currents; and bottom friction. Linearization is carried out on the time grid - nonlinear coefficients of the parabolic equation are taken for the previous time grid step. Next, a chain of tasks connected by the initial data - final solutions of the linearized mixed Cauchy problems chain on a uniform time grid is constructed, and thus, the linearization for the initial 2D nonlinear model is carried out in large. Earlier, the authors have proved the existence and uniqueness of the solution to a linear tasks chain. Prior estimate of the proximity of the linearized problem chain solution to the initial non-linear task solution has been also obtained. The conditions of its solution positivity and their convergence to the nonlinear sediment transport problem are defined in the norm of the Hilbert space L1 with the rate O(τ) where τ is a time step. Discussion and Conclusions. The obtained research results of the spatially two-dimensional nonlinear sediment transport model can be used for predicting the nonlinear hydrodynamic processes, improving their accuracy and reliability due to the availability of new accounting functionality of physically important factors, including the specification of the boundary conditions

Mathematical modeling of two-phase compressible fluid filtration based on modified adaptive method of minimum amendments

The work objective is to build and investigate the modified adaptive method of minimum amendments (MAMMA) which is destined for the numerical simulation of the two-phase compressible fluid filtration in porous media. This approach allows overcoming the known use limitations of other methods of the finite-difference equations solution, such as: crucial differential pressures acting on the oil-and-water bearing formation; and the compressibility of the medium at the considerable gas content in the oil phase. An approximation method - an explicit one for defining the function of water saturation, and an implicit one for the pressure function computation - is selected as the research basis. When setting the initial boundary value problem and its sampling, the process of the two-phase compressible fluid filtration in the space-dimensional domain with the lateral area bounded below by the subface of stratum, and above - by the bed top, is considered. A two-layer iterative method of the variational type - a modified method of minimal amendments adapted for solving finite-difference equations of the two-phase compressible fluid with a non-selfadjoint operator under the most general assumptions on the properties of the grid-problem operator is built. It is shown that a MAMMA has the asymptotic convergence rate characteristic of the “classical” alternate triangular method that does not use the Chebyshev acceleration technique and can be applied to the problems with a self-adjoint operator. Numerical experiments have confirmed the high efficiency of MAMMA. It is established that to achieve the specified accuracy, the number of iterations at the MAMMA reduces to 3-20 times as compared to the method of Seidel and the overrelaxation method

The difference scheme for the two-dimensional convection-diffusion problem for large peclet numbers

The purpose of this work is the development of a difference scheme for the solution of convection-diffusion problem at high Peclet numbers (Pe>2). In accordance with this purpose the following problems were solved: difference scheme for convection is built, comparison with the existing schemes is carried out; conditions for stability of the proposed difference scheme are obtained. Solutions of the convection-diffusion equation on the basis of the proposed difference scheme at various Peclet numbers are obtained

Correctness investigation for the suspension transport problem in coastal systems

This article is devoted to the confirmation the need for using a set of 3D dynamics models describing the various hydrophysical characteristics of the studied object to solve practical problems associated with the assessment of the ecological state of the water reservoirs. The present paper is devoted to the study of the three-dimensional model of transport and sedimentation of suspended matter in the coastal zone. The model takes into account such parameters as water movement, diffusionconvection, complicated bottom and shoreline geometry, lifting, transport and sedimentation of slurry. The existence and uniqueness of the solution of the corresponding indicated model of the initial-boundary value problem haas been envestigateded for two typical bottom boundary condirions. Also solution stability of the boundary-value problem in depend of functions: initial condition, boundary conditions and the righthand side in the norm L2 for any moment of time 0 < T < +∞, and also in the time-integral norm L2 has been proved. The model may be basis for the construction of hydrophysics models used to describe processes in the extraction of minerals from the seabed, in the dissemination of suspensions in shelf regions

Correctness investigation for the suspension transport problem in coastal systems

This article is devoted to the confirmation the need for using a set of 3D dynamics models describing the various hydrophysical characteristics of the studied object to solve practical problems associated with the assessment of the ecological state of the water reservoirs. The present paper is devoted to the study of the three-dimensional model of transport and sedimentation of suspended matter in the coastal zone. The model takes into account such parameters as water movement, diffusionconvection, complicated bottom and shoreline geometry, lifting, transport and sedimentation of slurry. The existence and uniqueness of the solution of the corresponding indicated model of the initial-boundary value problem haas been envestigateded for two typical bottom boundary condirions. Also solution stability of the boundary-value problem in depend of functions: initial condition, boundary conditions and the righthand side in the norm L2 for any moment of time 0 < T < +∞, and also in the time-integral norm L2 has been proved. The model may be basis for the construction of hydrophysics models used to describe processes in the extraction of minerals from the seabed, in the dissemination of suspensions in shelf regions

Practical Aspects of Implementation of the Parallel Algorithm for Solving Problem of Ctenophore Population Interaction in the Azov Sea

The paper covers the development and researching mathematical model of interaction processes between plankton and ctenophore populations based on the modern information technologies and computational methods, which leads to increase of the accuracy of predictive modeling of the ecology situation in shallow water in summer. The model takes into account the following: the transport of water environment; microturbulent diffusion; nonlinear interaction of plankton and ctenophore populations; biogenic, temperature and oxygen regimes; influence of salinity. The computational accuracy is significantly increased, and computational time is decreased at using the calculation method based on partially filled cells for discretization of model. The practical significance is the software implementation of the proposed model, the limits and prospects of its practical use are defined. Experimental software was developed based on multiprocessor computer system, which is intended for mathematical modeling of possible progress scenarios in shallow waters ecosystems on the example of the Azov Sea in summer. We used decomposition methods of grid domains in parallel implementation for computationally laborious convection-diffusion problems, taking into account the architecture and parameters of multiprocessor computer system.The paper covers the development and researching mathematical model of interaction processes between plankton and ctenophore populations based on the modern information technologies and computational methods, which leads to increase of the accuracy of predictive modeling of the ecology situation in shallow water in summer. The model takes into account the following: the transport of water environment; microturbulent diffusion; nonlinear interaction of plankton and ctenophore populations; biogenic, temperature and oxygen regimes; influence of salinity. The computational accuracy is significantly increased, and computational time is decreased at using the calculation method based on partially filled cells for discretization of model. The practical significance is the software implementation of the proposed model, the limits and prospects of its practical use are defined. Experimental software was developed based on multiprocessor computer system, which is intended for mathematical modeling of possible progress scenarios in shallow waters ecosystems on the example of the Azov Sea in summer. We used decomposition methods of grid domains in parallel implementation for computationally laborious convection-diffusion problems, taking into account the architecture and parameters of multiprocessor computer system