International Journal of Nonlinear Analysis and Applications
Comparative analysis of parallel algorithms for solving oil recovery problem using CUDA and OpenCL
Document Type : Research Paper
Authors
1 Yessenov University, Aktau, Kazakhstan
2 Al-Farabi Kazakh National University, Almaty, Kazakhstan
3 Al-Farabi Kazakh National University, Almaty, Kazakhstan
Abstract
In this paper the implementation of parallel algorithm of alternating direction implicit (ADI) method has been considered. ADI parallel algorithm is used to solve a multiphase multicomponent fluid flow problem in porous media. There are various technologies for implementing parallel algorithms on the CPU and GPU for solving hydrodynamic problems. In this paper GPU-based (graphic processor unit) algorithm was used. To implement the GPU-based parallel ADI method, CUDA and OpenCL were used. ADI is an iterative method used to solve matrix equations. To solve the tridiagonal system of equations in ADI method, the parallel version of cyclic reduction (CR) method was implemented. The cyclic reduction is a method for solving linear equations by repeatedly splitting a problem as a Thomas method. To implement of a sequential algorithm for solving the oil recovery problem, the implicit Thomas method was used. Thomas method or tridiagonal matrix algorithm is used to solve tridiagonal systems of equations. To test parallel algorithms personal computer installed Nvidia RTX 2080 graphic card with 8 GB of video memory was used. The computing results of parallel algorithms using CUDA and OpenCL were compared and analyzed. The main purpose of this research work is a comparative analysis of the parallel algorithm computing results on different technologies, in order to show the advantages and disadvantages each of CUDA and OpenCL for solving oil recovery problems.
Keywords
[2] D.Zh. Akhmed-Zaki, T.S. Imankulov, B. Matkerim, B.S. Daribayev, K.A. Aidarov and O.N. Turar, Large-scale simulation of oil recovery by surfactant-polymer flooding, Eurasian J. Math. Comput. Appl. 4 (2016) 12–31.
[3] R. Banger and K. Bhattacharyya, OpenCL Programming by Example, Packt Publishing, 2013.
[4] J. Bear, Dynamics of Fluids in Porous Media, New York: Dover, 1972.
[5] V. Casulli and R. Cheng, Semi-implicit finite difference methods for three-dimensional shallow water flow, Int. J. Numerical Meth. Fluids 15 (2005) 629–648.
[6] Z. Chen, Reservoir Simulation: Mathematical Techniques in Oil Recovery, Society for Industrial and Applied Mathematics, 2007.
[7] J. Cheng, M. Grossman and T. McKercher, Professional CUDA C Programming, Wrox, 1st edition, 2014.
[8] Sh. Cook, CUDA Programming: A Developer’s Guide to Parallel Computing with GPUs (Applications of GPU Computing), 1st edition, Morgan Kaufmann, 2012.
[9] S. Cook, CUDA Programming: A Developer’s Guide to Parallel Computing with GPUs (Applications of GPU Computing), Morgan Kaufmann, 1st edition, 2012.
[10] A. Davidson and J. Owens, Register packing for cyclic reduction: a case study, Proc. Fourth Workshop on General Purpose Processing on Graphics Processing Units, ACM, 2011.
[11] A. Davidson, Y. Zhang and J. Owens, An auto-tuned method for solving large tridiagonal systems on the GPU, Parallel & Distributed Processing Symposium (IPDPS), IEEE Int. (2011) 956–965.
[12] N. Ellner and E. Wachspress, Alternating direction implicit iteration for systems with complex spectra, SIAM J. Numerical Anal. 28 (1991) 859–870.
[13] D. Goddeke and R. Strzodka, Cyclic reduction tridiagonal solvers on GPUs applied to mixed-precision multigrid, Parallel Distributed Syst. IEEE Trans. 22(1) (2011) 22–32.
[14] J. Han and B. Sharma, Learn CUDA Programming: A beginner’s guide to GPU programming and parallel computing with CUDA 10.x and C/C++, Packt Publishing, 2019.
[15] N. Hecquet, Parallel Calculus in CUDA: Heat equation, Black and Scholes Model, Editions universitaires eu- ´rop´eennes, 2019.
[16] T.S. Imankulov, D.Zh. Akhmed-Zaki, B.S. Daribayev and et al., Intellectual system for analysing thermal compositional modelling with chemical reactions, 16th European Conference on the Mathematics of Oil Recovery,2018.
[17] T.S. Imankulov, D.Zh. Akhmed-Zaki, B.S. Daribayev and O.N. Turar, HPC Mobile Platform for Solving Oil Recovery Problem, Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2016). 2 (2016) 595–598.
[18] B. Jang, D. Kaeli, S. Do, H. Pien, Multi GPU implementation of iterative tomographic reconstruction algorithms, IEEE Int. Symp. Biomedical Imaging: From Nano to Macro. (2009) 185–188.
[20] M. Kass, A. Lefohn, J.D. Owens, Interactive depth of field using simulated diffusion, Technical Report 06-01, Pixar Animation Studios, 2006.
[21] M. Kass, G. Miller, Rapid, stable fluid dynamics for computer graphics, Computer Graphics (Proceedings of SIGGRAPH 90), (1990) 49—57.
[22] H. Kim, S. Wu, L. Chang, W. Hwu, A scalable tridiagonal solver for GPUs, Parallel Processing (ICPP), Int. Conf. IEEE. (2011) 444–453.
[23] J. Kowalik, T. Puzniakowski, Using OpenCL: Programming Massively Parallel Computers (Advances in Parallel Computing), IOS Press, Har/Cdr edition, 2012.
[24] I. Lindemuth, J. Killeen, Alternating direction implicit techniques for two-dimensional magnetohydrodynamic calculations, Journal of Computational Physics. 13 (1972) 181–208.
[25] J.J.Martin, Cubic Equations of State, EC Fundamentals, 1979.
[26] A. Munshi, OpenCL Programming Guide, Addison-Wesley Professional, 1st edition, 2011.
[27] T. Namiki, A new FDTD algorithm based on alternating-direction implicit method, Microwave Theory and Techniques, IEEE Transactions on 47. (1999) 2003–2007.
[28] D. Peaceman and H. Rachford, The numerical solution of parabolic and elliptic differential equations, Journal of the Society for Industrial & Applied Mathematics. 3 (1955) 28–41.
[29] K.S.Pederson and P.L. Christensen, Phase Behavior of Petroleum Reservoir Fluids, CRC Press, 2008.
[30] D.Y. Peng and D.B. Robinson, A new two-constant equation of state, Industrial and Engineering Chemistry Fundamentals. 15 (1976) 59—64.
[31] R.C. Reid, J.M. Prausnitz and T.K. Sherwood, The Properties of Gases and Liquids 3rd edition, New York: McGraw-Hill, 1977.
[32] N. Sakharnykh, Effcient tridiagonal solvers for ADI methods and fluid simulation, NVIDIA GPU Technology Conference, 2010.
[33] N. Sakharnykh, Tridiagonal solvers on the GPU and applications to fluid simulation, GPU Technology Conference, 2009.
[34] J. Sanders and E. Kandrot, CUDA by Example: An Introduction to General-Purpose GPU Programming, AddisonWesley Professional, 1st edition, 2010.
[35] S. Sengupta, M. Harris, Y. Zhang and J.D. Owens, Scan primitives for GPU computing, Graphics Hardware, ACM, New York. (2007) 97-–106.
[36] G. Soave, Equilibrium constants from a modified Redlich-Kwong equation of state, Chem. Eng. Sci. 27 (1972) 1197—1203.
[37] D. Storti and M. Yurtoglu, CUDA for Engineers: An Introduction to High-Performance Parallel Computing, Addison-Wesley Professional, 1st edition, 2015.
[38] R. Tay, OpenCL Parallel Programming Development Cookbook, Packt Publishing. 2013.
[39] L.H. Thomas, Elliptic Problems in Linear Differential Equations over a Network, Watson Sci. Comput. Lab Report. 1949.
[40] W. Wu, Computational river dynamics, CRC. 2007.
[41] Y. Zhang, J. Cohen, A. Davidson and J. Owens, A hybrid method for solving tridiagonal systems on the GPU, GPU Computing Gems Jade Edition, 2011.
[42] Y. Zhang, J. Cohen and J. Owens, Fast tridiagonal solvers on the GPU, ACM Sigplan Notices 45 (2010) 127–136.
[43] Y. Zhang and Y. Jia, Parallelization of implicit CCHE2D model using CUDA programming techniques, World Envir. Water Resour. Cong. (2013) 1777–1792.
[44] Y. Zhang and Y. Jia, Parallelized CCHE2D model with CUDA Fortran on graphics process units, Comput. Fluids 2013 (2013) 359–368.
[45] D. Zudkevitch and J. Joffe, Correlation and prediction of vapor-liquid equilibria with the Redlich-Kwong equation of state, Amer. Inst. Chem. Engin. J. 16 (1970) 112—199.
)
Volume 12, Issue 1
May 2021Pages 351-364
- Receive Date: 10 October 2020
- Revise Date: 25 January 2021
- Accept Date: 27 January 2021