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A graphics processing unit-based robust numerical model for solute transport driven by torrential flow condition

基于图形处理器加速的急变流条件下溶质输移的稳健数值模型

Abstract

Solute transport simulations are important in water pollution events. This paper introduces a finite volume Godunov-type model for solving a 4×4 matrix form of the hyperbolic conservation laws consisting of 2D shallow water equations and transport equations. The model adopts the Harten-Lax-van Leer-contact (HLLC)-approximate Riemann solution to calculate the cell interface fluxes. It can deal well with the changes in the dry and wet interfaces in an actual complex terrain, and it has a strong shock-wave capturing ability. Using monotonic upstream-centred scheme for conservation laws (MUSCL) linear reconstruction with finite slope and the Runge-Kutta time integration method can achieve second-order accuracy. At the same time, the introduction of graphics processing unit (GPU)-accelerated computing technology greatly increases the computing speed. The model is validated against multiple benchmarks, and the results are in good agreement with analytical solutions and other published numerical predictions. The third test case uses the GPU and central processing unit (CPU) calculation models which take 3.865 s and 13.865 s, respectively, indicating that the GPU calculation model can increase the calculation speed by 3.6 times. In the fourth test case, comparing the numerical model calculated by GPU with the traditional numerical model calculated by CPU, the calculation efficiencies of the numerical model calculated by GPU under different resolution grids are 9.8–44.6 times higher than those by CPU. Therefore, it has better potential than previous models for large-scale simulation of solute transport in water pollution incidents. It can provide a reliable theoretical basis and strong data support in the rapid assessment and early warning of water pollution accidents.

概要

目的

暴雨山洪灾害会对人类的生命安全和经济活动产生巨大影响. 此类洪水事件会破坏化工厂或污水处理厂等可能释放有害溶质的设施, 使释放的溶质随洪水向洪泛区或地势低洼处输移, 进而严重影响公共卫生安全, 加剧洪水对人类造成的危害. 因此, 需要一个高效稳健的数值模型来对其进行快速预警和评估.

创新点

1. 提出了一种基于图形处理器(GPU)加速的急变流驱动溶质运移的稳健数值模型; 2. 探讨不同型号GPU和中央处理机(CPU)的计算性能和加速比.

方法

1. 采用Godunov格式的有限体积法求解二维浅水方程和溶质输移方程, 利用HLLC近似黎曼求解器计算单元网格界面通量, 并应用MUSCL限坡线性重建和龙格-库塔时间积分法实现二阶精度. 2. 引入GPU加速计算技术提高模型计算效率.

结论

1. 通过理想算例和经典算例对模型精度和稳定性的验证, 表明该模型能够有效地抑制数值阻尼和虚假的数值振荡, 并且具有较好的和谐性; 2. 采用不同型号的GPU和CPU计算模型模拟相同的事件, 表明GPU加速技术在保证模拟精度的同时可实现大规模高效率计算; 3. 该模型能够快速准确地模拟暴雨山洪或溃坝洪水引起的大规模突然性溶质输移过程, 可以为水污染事故提供可靠的理论依据和有力的数据支撑.

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References

  1. Aidun CK, Clausen JR, 2010. Lattice-Boltzmann method for complex flows. Annual Review of Fluid Mechanics, 42(1):439–472. https://doi.org/10.1146/annurev-fluid-121108-145519

    MathSciNet  MATH  Article  Google Scholar 

  2. Audusse E, Bristeau MO, 2003. Transport of pollutant in shallow water a two time steps kinetic method. ESAIM: M2AN, 37(2):389–416. https://doi.org/10.1051/m2an:2003034

    MathSciNet  MATH  Article  Google Scholar 

  3. Barredo JI, 2007. Major flood disasters in Europe: 1950–2005. Natural Hazards, 42(1):125–148. https://doi.org/10.1007/s11069-006-9065-2

    Article  Google Scholar 

  4. Bayazıt Y, Koç C, Bakış R, 2021. Urbanization impacts on flash urban floods in Bodrum Province, Turkey. Hydrological Sciences Journal, 66(1):118–133. https://doi.org/10.1080/02626667.2020.1851031

    Article  Google Scholar 

  5. Begnudelli L, Sanders BF, 2006. Unstructured grid finite-volume algorithm for shallow-water flow and scalar transport with wetting and drying. Journal of Hydraulic Engineering, 132(4):371–384. https://doi.org/10.1061/(asce)0733-9429(2006)132:4(371)

    Article  Google Scholar 

  6. Benkhaldoun F, Elmahi I, Seaïd M, 2007. Well-balanced finite volume schemes for pollutant transport by shallow water equations on unstructured meshes. Journal of Computational Physics, 226(1):180–203. https://doi.org/10.1016/j.jcp.2007.04.005

    MathSciNet  MATH  Article  Google Scholar 

  7. Bi S, Zhou JZ, Cheng SS, et al., 2013. A high-precision two-dimensional flow-transport coupled model based on Godunov’s schemes. Advances in Water Science, 24(5): 706–714 (in Chinese). https://doi.org/10.14042/j.cnki.32.1309.2013.05.003

    Google Scholar 

  8. Cao Y, Ye YT, Liang LL, et al., 2019. High efficient and accurate simulation of pollutant transport in torrential flow based on adaptive grid method. Journal of Hydraulic Engineering, 50(3):388–398 (in Chinese). https://doi.org/10.13243/j.cnki.slxb.20180891

    Google Scholar 

  9. Chen CY, Lu TH, Yang YF, et al., 2021. Toxicokinetic/toxicodynamic-based risk assessment of freshwater fish health posed by microplastics at environmentally relevant concentrations. Science of the Total Environment, 756: 144013. https://doi.org/10.1016/j.scitotenv.2020.144013

    Article  Google Scholar 

  10. Dong J, 2020. A robust central scheme for the shallow water flows with an abrupt topography based on modified hydrostatic reconstructions. Mathematical Methods in the Applied Sciences, 43(15):9024–9045. https://doi.org/10.1002/mma.6597

    MathSciNet  MATH  Article  Google Scholar 

  11. Duarte HDO, Droguett EL, Araújo M, et al., 2013. Quantitative ecological risk assessment of industrial accidents: the case of oil ship transportation in the coastal tropical area of northeastern Brazil. Human and Ecological Risk Assessment: An International Journal, 19(6):1457–1476. https://doi.org/10.1080/10807039.2012.723187

    Article  Google Scholar 

  12. Fraccarollo L, Toro EF, 1995. Experimental and numerical assessment of the shallow water model for two-dimensional dam-break type problems. Journal of Hydraulic Research, 33(6):843–864. https://doi.org/10.1080/00221689509498555

    Article  Google Scholar 

  13. Harten A, Lax PD, van Leer B, 1983. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Review, 25(1):35–61. https://doi.org/10.1137/1025002

    MathSciNet  MATH  Article  Google Scholar 

  14. Hou JM, Simons F, Hinkelmann R, 2012. Improved total variation diminishing schemes for advection simulation on arbitrary grids. International Journal for Numerical Methods in Fluids, 70(3):359–382. https://doi.org/10.1002/fld.2700

    MathSciNet  MATH  Article  Google Scholar 

  15. Hou JM, Simons F, Hinkelmann R, 2013a. A new TVD method for advection simulation on 2D unstructured grids. International Journal for Numerical Methods in Fluids, 71(10):1260–1281. https://doi.org/10.1002/fld.3709

    MathSciNet  MATH  Article  Google Scholar 

  16. Hou JM, Simons F, Mahgoub M, et al., 2013b. A robust well-balanced model on unstructured grids for shallow water flows with wetting and drying over complex topography. Computer Methods in Applied Mechanics and Engineering, 257:126–149. https://doi.org/10.1016/j.cma.2013.01.015

    MathSciNet  MATH  Article  Google Scholar 

  17. Hou JM, Liang QH, Zhang HB, et al., 2014. Multislope MUSCL method applied to solve shallow water equations. Computers & Mathematics with Applications, 68(12): 2012–2027. https://doi.org/10.1016/j.camwa.2014.09.018

    MathSciNet  MATH  Article  Google Scholar 

  18. Hou JM, Liang QH, Li ZB, et al., 2015. Numerical error control for second-order explicit TVD scheme with limiters in advection simulation. Computers & Mathematics with Applications, 70(9):2197–2209. https://doi.org/10.1016/j.camwa.2015.08.022

    MathSciNet  MATH  Article  Google Scholar 

  19. Hou JM, Wang R, Liang QH, et al., 2018a. Efficient surface water flow simulation on static Cartesian grid with local refinement according to key topographic features. Computers & Fluids, 176:117–134. https://doi.org/10.1016/j.compfluid.2018.03.024

    MathSciNet  MATH  Article  Google Scholar 

  20. Hou JM, Wang T, Li P, et al., 2018b. An implicit friction source term treatment for overland flow simulation using shallow water flow model. Journal of Hydrology, 564:357–366. https://doi.org/10.1016/j.jhydrol.2018.07.027

    Article  Google Scholar 

  21. Hou JM, Kang YD, Hu CH, et al., 2020. A GPU-based numerical model coupling hydrodynamical and morphological processes. International Journal of Sediment Research, 35(4):386–394. https://doi.org/10.1016/j.ijsrc.2020.02.005

    Article  Google Scholar 

  22. Kachiashvili K, Gordeziani D, Lazarov R, et al., 2007. Modeling and simulation of pollutants transport in rivers. Applied Mathematical Modelling, 31(7):1371–1396. https://doi.org/10.1016/j.apm.2006.02.015

    MATH  Article  Google Scholar 

  23. Kawahara M, Umetsu T, 1986. Finite element method for moving boundary problems in river flow. International Journal for Numerical Methods in Fluids, 6(6):365–386. https://doi.org/10.1002/fld.1650060605

    MATH  Article  Google Scholar 

  24. Kong J, Xin P, Shen CJ, et al., 2013. A high-resolution method for the depth-integrated solute transport equation based on an unstructured mesh. Environmental Modelling & Software, 40:109–127. https://doi.org/10.1016/j.envsoft.2012.08.009

    Article  Google Scholar 

  25. La Rocca M, Montessori A, Prestininzi P, et al., 2015. A multispeed discrete Boltzmann model for transcritical 2D shallow water flows. Journal of Computational Physics, 284:117–132. https://doi.org/10.1016/j.jcp.2014.12.029

    MathSciNet  MATH  Article  Google Scholar 

  26. La Rocca M, Miliani S, Prestininzi P, 2020. Discrete Boltzmann numerical simulation of simplified urban flooding configurations caused by dam break. Frontiers in Earth Science, 8:346. https://doi.org/10.3389/feart.2020.00346

    Article  Google Scholar 

  27. Li YP, Wei J, Gao XM, et al., 2018. Turbulent bursting and sediment resuspension in hyper-eutrophic Lake Taihu, China. Journal of Hydrology, 565:581–588. https://doi.org/10.1016/j.jhydrol.2018.08.067

    Article  Google Scholar 

  28. Liang QH, 2010. A well-balanced and non-negative numerical scheme for solving the integrated shallow water and solute transport equations. Communications in Computational Physics, 7(5):1049–1075. https://doi.org/10.4208/cicp.2009.09.156

    MathSciNet  MATH  Article  Google Scholar 

  29. Liang QH, Xia XL, Hou JM, 2016. Catchment-scale highresolution flash flood simulation using the GPU-based technology. Procedia Engineering, 154:975–981. https://doi.org/10.1016/j.proeng.2016.07.585

    Article  Google Scholar 

  30. Liu RX, 2011. Simulation of water transferring impact on the water quality in Danjiangkou reservoir of the south to north water diversion middle line project. Journal of Basic Science and Engineering, 19(S1):193–200 (in Chinese). https://doi.org/10.3969/j.issn.1005-0930.2011.s1.022

    Google Scholar 

  31. Liu WH, Chen RQ, Qiu RF, et al., 2020. Generalized form of interpolation-supplemented lattice Boltzmann method for computational aeroacoustics. Journal of Zhejiang University (Engineering Science), 54(8):1637–1644 (in Chinese). https://doi.org/10.3785/j.issn.1008-973X.2020.08.024

    Google Scholar 

  32. Murillo J, García-Navarro P, Burguete J, 2009. Conservative numerical simulation of multi-component transport in two-dimensional unsteady shallow water flow. Journal of Computational Physics, 228(15):5539–5573. https://doi.org/10.1016/j.jcp.2009.04.039

    MathSciNet  MATH  Article  Google Scholar 

  33. Petti M, Bosa S, 2007. Accurate shock-capturing finite volume method for advection-dominated flow and pollution transport. Computers & Fluids, 36(2):455–466. https://doi.org/10.1016/j.compfluid.2005.11.008

    MATH  Article  Google Scholar 

  34. Roccati A, Faccini F, Luino F, et al., 2019. Heavy rainfall triggering shallow landslides: a susceptibility assessment by a GIS-approach in a Ligurian Apennine catchment (Italy). Water, 11(3):605. https://doi.org/10.3390/w11030605

    Article  Google Scholar 

  35. Shao JR, Wu SQ, Zhou J, et al., 2012. High-accuracy numerical simulation of 2D transport problems. Advances in Water Science, 23(3):383–389 (in Chinese). https://doi.org/10.14042/j.cnki.32.1309.2012.03.008

    Google Scholar 

  36. Smith LS, Liang QH, 2013. Towards a generalised GPU/CPU shallow-flow modelling tool. Computers & Fluids, 88: 334–343. https://doi.org/10.1016/j.compfluid.2013.09.018

    MathSciNet  MATH  Article  Google Scholar 

  37. Song LX, Yang F, Hu XZ, et al., 2014. A coupled mathematical model for two-dimensional flow-transport simulation in tidal river network. Advances in Water Science, 25(4): 550–559 (in Chinese). https://doi.org/10.14042/j.cnki.32.1309.2014.04.012

    Google Scholar 

  38. Tang CY, Li YP, Jiang P, et al., 2015. A coupled modeling approach to predict water quality in Lake Taihu, China: linkage to climate change projections. Journal of Freshwater Ecology, 30(1):59–73. https://doi.org/10.1080/02705060.2014.999360

    Article  Google Scholar 

  39. Tao T, Lu YJ, Fu X, et al., 2012. Identification of sources of pollution and contamination in water distribution networks based on pattern recognition. Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering), 13(7):559–570. https://doi.org/10.1631/jzus.A1100286

    Article  Google Scholar 

  40. Toro EF, 2001. Shock-capturing Methods for Free-surface Shallow Flow. John Wiley & Sons, Ltd., Chichester, UK.

    MATH  Google Scholar 

  41. Venturi S, Francesco SD, Geier M, et al., 2021. Modelling flood events with a cumulant CO lattice Boltzmann shallow water model. Natural Hazards, 105(2):1815–1834. https://doi.org/10.1007/s11069-020-04378-x

    Article  Google Scholar 

  42. Wang ZW, Cheng WP, 2002. Analysis of ecological mechanism of urban flood and waterlog—research based mainly on Hangzhou City. Journal of Zhejiang University (Engineering Science), 36(5):582–587 (in Chinese). https://doi.org/10.3785/jissn.1008-973X.2002.05.023

    Google Scholar 

  43. Zhang LL, Liang QH, Wang YL, et al., 2015. A robust coupled model for solute transport driven by severe flow conditions. Journal of Hydro-environment Research, 9(1):49–60. https://doi.org/10.1016/j.jher.2014.04.005

    Article  Google Scholar 

  44. Zhang WJ, Lin XY, Su XS, 2010. Transport and fate modeling of nitrobenzene in groundwater after the Songhua River pollution accident. Journal of Environmental Management, 91(11):2378–2384. https://doi.org/10.1016/j.jenvman.2010.06.025

    Article  Google Scholar 

  45. Zhang WW, 2011. Measuring the value of water quality improvements in Lake Tai, China. Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering), 12(9):710–719. https://doi.org/10.1631/jzus.A11b0157

    Article  Google Scholar 

  46. Zhao Y, Nan J, Cui FY, et al., 2007. Water quality forecast through application of BP neural network at Yuqiao reservoir. Journal of Zhejiang University-SCIENCE A, 8(9): 1482–1487. https://doi.org/10.1631/jzus.2007.A1482

    Article  Google Scholar 

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Author information

Affiliations

Authors

Corresponding author

Correspondence to Bao-shan Shi.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 52009104 and 52079106), the Shaanxi Provincial Department of Water Resources Project (No. 2017slkj-14), and the Shaanxi Provincial Department of Science and Technology Project (No. 2017JQ3043), China

Contributors

Jing-ming HOU and Qiu-hua LIANG developed the numerical model of the GPU-accelerated surface water flow and associated transport. Bao-shan SHI perfected the model and wrote the manuscript. Yu TONG, Yong-de KANG, Zhao-an ZHANG, and Gang-gang BAI helped to organize the manuscript. Xu-jun GAO and Xiao YANG revised and edited the final version. Jing-ming HOU and Bao-shan SHI contributed equally to this work.

Conflict of interest

Jing-ming HOU, Bao-shan SHI, Qiu-hua LIANG, Yu TONG, Yong-de KANG, Zhao-an ZHANG, Gang-gang BAI, Xu-jun GAO, and Xiao YANG declare that they have no conflict of interest.

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Cite this article

Hou, Jm., Shi, Bs., Liang, Qh. et al. A graphics processing unit-based robust numerical model for solute transport driven by torrential flow condition. J. Zhejiang Univ. Sci. A 22, 835–850 (2021). https://doi.org/10.1631/jzus.A2000585

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Key words

  • Solute transport
  • Shallow water equations
  • Godunov-type scheme
  • Harten-Lax-van Leer-contact (HLLC) Riemann solver
  • Graphics processing unit (GPU) acceleration technology
  • Torrential flow

CLC number

  • TV131.2

关键词

  • 溶质输移
  • 浅水方程
  • Godunov格式
  • HLLC黎曼求解器
  • GPU加速
  • 急变流