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Numerical simulation based on fine-scale reservoir models helps petroleum engineers in understanding fluid flow in porous media and achieving higher recovery ratio. Fine-scale models give rise to large-scale linear systems, and thus require effective solvers for solving these linear systems to finish simulation in reasonable turn-around time. In this paper, we study convergence, robustness, and efficiency of a class of multi-stage preconditioners accelerated by Krylov subspace methods for solving Jacobian systems from a fully implicit discretization. We compare components of these preconditioners, including decoupling and sub-problem solvers, for fine-scale reservoir simulation. Several benchmark and real-world problems, including a ten-million-cell reservoir problem, were simulated on a desktop computer. Numerical tests show that the combination of the alternating block factorization method and multi-stage subspace correction preconditioner gives a robust and memory-efficient solver for fine-scale reservoir simulation.

Little seems to be known about the study of the chaotic system with only Lyapunov stable equilibria from the perspective of differential geometry. Therefore, this paper presents Jacobi analysis of an unusual three-dimensional (3D) autonomous chaotic system. Under certain parameter conditions, this system has positive Lyapunov exponents and only two linear stable equilibrium points, which means that chaotic attractor and Lyapunov stable equilibria coexist. The dynamical behavior of the deviation vector near the whole trajectories (including all equilibrium points) is analyzed in detail. The results show that the value of the deviation curvature tensor at equilibrium points is only related to parameters; the two equilibrium points of the system are Jacobi stable if the parameters satisfy certain conditions. Particularly, for a specific set of parameters, the linear stable equilibrium points of the system are always Jacobi unstable. A periodic orbit that is Lyapunov stable is also proven to be always Jacobi unstable. Next, Jacobi-stable regions of the Lorenz system, the Chen system and the system under study are compared for specific parameters. It can be found that although these three chaotic systems are very similar, their regions of Jacobi stable parameters are much different. Finally, by comparing Jacobi stability with Lyapunov stability, the obtained results demonstrate that the Jacobi stable parameter region is basically symmetric with the Lyapunov stable parameter region.

In this study, parallelization of a Discrete Element Method (DEM) code titled Trubal was carried out based on the CPU–GPU heterogeneous architecture where both two- and three-dimensional cases were assessed. In Trubal, the particle–particle/wall interaction rules are governed by the theoretical contact mechanics which enable the direct use of real physical material properties in the calculation. We reconstructed Trubal in two steps: reconstruction of the static storage structure; essential parallelism on the relative newer version code. Numerical simulations were implemented to present the benefits of this research. Firstly, two simulations of die filling with a moving shoe involving 6000 and 60,000 two-dimensional particles were conducted under (i) NVIDIA Tesla C2050 card together with Intel Core-Duo 2.93 GHz CPU and (ii) NVIDIA Tesla K40c card along with Intel Xeon 3.00 GHz CPU. Average speedups of (i) 4.69 and 12.78 as well as (ii) 6.52 and 18.60 in computational time were obtained, respectively. Then, a simulation of die filling with a stationary shoe containing 20,000 three-dimensional particles was carried out under the same conditions where average speedups of (i) 12.90 and (ii) 19.66 in computational time were obtained, respectively. It is shown that the final version parallel code gave a substantial acceleration on the original Trubal.

As a result of the interplay between advances in computer hardware, software, and algorithm, we are now in a new era of large-scale reservoir simulation, which focuses on accurate flow description, fine reservoir characterization, efficient nonlinear/linear solvers, and parallel implementation. In this paper, we discuss a multilevel preconditioner in a new-generation simulator and its implementation on multicore computers. This preconditioner relies on the method of subspace corrections to solve large-scale linear systems arising from fully implicit methods in reservoir simulations. We investigate the parallel efficiency and robustness of the proposed method by applying it to million-cell benchmark problems.

The geometric multigrid method (GMG) is one of the most efficient solving techniques for discrete algebraic systems arising from elliptic partial differential equations. GMG utilizes a hierarchy of grids or discretizations and reduces the error at a number of frequencies simultaneously. Graphics processing units (GPUs) have recently burst onto the scientific computing scene as a technology that has yielded substantial performance and energy-efficiency improvements. A central challenge in implementing GMG on GPUs, though, is that computational work on coarse levels cannot fully utilize the capacity of a GPU. In this work, we perform numerical studies of GMG on CPU-GPU heterogeneous computers. Furthermore, we compare our implementation with an efficient CPU implementation of GMG and with the most popular fast Poisson solver, Fast Fourier Transform, in the cuFFT library developed by NVIDIA.

Incomplete LU (ILU) methods are widely used in petroleum reservoir simulation and many other applications. However high complexity often makes them the hotspot in the whole simulation due to high complexity when problem size is large. ILU’s inherent serial nature also makes them difficult to take full advantage of computing power of multi-core and many-core devices. In this paper, a greedy graph coloring method is applied to the ILU(k) factorization and triangular solution phases. This method increases degree of parallelism and improves load balance. A block-wise storage format is employed in our ILU implementation in order to take advantage of hierarchical memory structures. Moreover, a dual intensive parallel model is proposed to further improve the performance of ILU(k) on GPUs. We test the performance of the proposed parallel ILU(k) with a set of Jacobian systems arising from petroleum reservoir simulation. Numerical results suggest that the proposed parallel ILU(k) method is effective and robust on multi-core and many-core architectures. On an Intel Xeon E5 multi-core CPU, the speedup compared with the serial execution time is \(5.6\times \) and \(5.4\times \) for factorization and triangular solution, respectively; on an Nvidia K40c GPU card, the speedup can reach \(8.6\times \) and \(12.7\times \) for factorization and triangular solution, respectively.

Parallel reservoir simulators are now widely used with availability of super computers. Modern massively parallel supercomputers demonstrate great power for simulating large-scale reservoir models. However, improving scalability and efficiency for fully implicit methods on emerging parallel architectures is still challenging. In this paper, we present a robust discretization together with a parallel linear solver algorithm; and we explore the parallel implementation on the world’s fastest supercomputer Tianhe-2. Starting with a general compositional model, we focus on the black oil model and developed Parallel eXtension Framework for parallelizing the serial simulator. A parallel preconditioner based on fast auxiliary space preconditioning (FASP) is applied to solve the Jacobian system arising from the fully implicit discretization. The parallel simulator was validated using large-scale black oil benchmark problems, for which parallel scalabilities were tested. Giant reservoir models with over 100 million grid blocks have been simulated within a few minutes, and test the strong scalability of AMG solver with 1 billion unknown. We also demonstrate the parallelization and acceleration using Intel Xeon Phi coprocessors. In the end, the efficiency of the parallel simulator is illustrated by a giant reservoir using up to 10,000 cores, for which the CPU and communication time are summarized for the linear and nonlinear algorithms.

In this paper, we propose a preconditioner for solving large-scale linear systems arising from fully implicit petroleum reservoir simulation. First, we take several analytical and physical considerations into account, and choose appropriate auxiliary spaces (problems) to design a preconditioner in the framework of auxiliary space preconditioning method. Secondly, we present an efficient implementation of the proposed preconditioner in modern multicore computer environment. Finally, we test the efficiency and robustness of the proposed methods with a large benchmark problem.

In this paper, we study several unsmoothed aggregation based algebraic multigrid (UA-AMG) methods with regard to different characteristics of CPUs and graphics processing units (GPUs). We propose some UA-AMG methods with lower computational complexity for CPU and CPU-GPU, and study these UA-AMG methods mixing with 4 kinds of red-black colored Gauss-Seidel smoothers for CPU-GPU since the initial mesh is structured. These UA-AMG methods are used as preconditioners for the conjugate gradient (CG) solver to solve a class of two-dimensional single-temperature radiation diffusion equations discretized by preserving symmetry finite volume element scheme. Numerical results demonstrate that, UA-NA-CG-s, which wins the best robustness and efficiency among them, is much more efficient than the default AMG preconditioned CG solvers in HYPRE, AGMG and Cusp for CPU; Under CPU-GPU, UA-W-CG-p is the most robust and efficient one, and rather more efficient than the smoothed aggregation based AMG preconditioned CG solver in Cusp.Copyright © 2013 by ASME

Algebraic multigrid(AMG) method is one of the most efficient iterative methods for solving the linear systems which arising from discretizations of partial differential equations. Parallel AMG has been widely used in large-scale scientific and engineering computation. In this paper, considering a class of linear algebraic equations with sparse and banded coefficient matrices, we improve the OpenMP version of BoomerAMG by modifying its modules of the parallel interpolation and the parallel coarse grid operator. The improved version of BoomerAMG is applied to solve the Laplace equation and a class of two-dimensional three-temperature radiative diffusion equations. Numerical results demonstrate that the new method yields better scalability and efficiency.

In this paper, we propose two parallel Hilbert Space-filling Curve(HSFC) generation algorithms BMIMp and SDDMp based on block matrix iteration method(BMIM) and state diagrams driver method(SDDM) in the CUDA parallel programming mode. Numerical results show that both of them obtain high parallel speedup. Especially, the speedup of BMIMp and SDDMp can reach 207 and 290 respectively for the 14-order HSFC. Furthermore, BMIMp outperforms SDDMp when considering the total computation time.