A new high-order particle method for solving high Reynolds number incompressible flows

  • Rex Kuan-Shuo Liu
  • Khai-Ching Ng
  • Tony Wen-Hann SheuEmail author


In this study, a new high-order particle method is proposed to solve the incompressible Navier–Stokes equations. The proposed method combines the advantages of particle and mesh methods to approximate the total and the spatial derivative terms under the Lagrangian and the Eulerian frameworks. Our aim is to avoid convective instability and increase solution accuracy at the same time. Data transfer from Lagrangian particles to Eulerian grids is realized by moving least squares interpolation. In contrast to the previously proposed method, there is no need to interpolate diffusion terms from Eulerian grids to Lagrangian particles. Therefore, the accuracy of the present solution will not be deteriorated by interpolation error. Additionally, no extra work is required to manage particles for searching procedure. Because no convection term needs to be discretized by upwinding schemes, false diffusion and dispersion errors will not be introduced, thereby increasing the solution accuracy. To verify the proposed particle method, several benchmark problems are solved to show that the present simulation is more stable, accurate, and efficient. The proposed particle method renders fourth- and second-order accurate solutions in space for velocity and pressure, respectively.


High-order particle method Incompressible Navier–Stokes equations False diffusion error Numerical dispersion error 



This work was supported by the Ministry of Science and Technology (MOST) of Republic of China (R.O.C.) under the Grants MOST 106-2221-E-002-107-MY2.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest.


  1. 1.
    Koshizuka S, Oka Y (1996) Moving-particle semi-implicit method for fragmentation of incompressible fluid. Nucl Sci Eng 123:421–434CrossRefGoogle Scholar
  2. 2.
    Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon Not R Astron Soc 181:375–389CrossRefzbMATHGoogle Scholar
  3. 3.
    Lucy LB (1977) A numerical approach to the testing of the fission hypothesis. Astron J 82:1013–1024CrossRefGoogle Scholar
  4. 4.
    Hwang YH (2011) A moving particle method with embedded pressure mesh (MPPM) for incompressible flow calculations. Numer Heat Transf B Fundam 60:370–398CrossRefGoogle Scholar
  5. 5.
    Shibata K, Masaie I, Kondo M, Murotani K, Koshizuka S (2015) Improved pressure calculation for the moving particle semi-implicit method. Comput Part Mech 2:91–108CrossRefGoogle Scholar
  6. 6.
    Xu T, Kin YC (2016) Improvements for accuracy and stability in a weakly-compressible particle method. Comput Fluids 137:1–14MathSciNetCrossRefGoogle Scholar
  7. 7.
    Sanchez-Mondragon J (2016) On the stabilization of unphysical pressure oscillations in MPS method simulations. Int J Numer Methods Fluids 82:471–492MathSciNetCrossRefGoogle Scholar
  8. 8.
    Tanaka M, Cardoso R, Bahai H (2018) Multi-resolution MPS method. J Comput Phys 359:106–136MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Park S, Jeun G (2011) Coupling of rigid body dynamics and moving particle semi-implicit method for simulating isothermal multi-phase fluid interactions. Comput Methods Appl Mech Eng 200:130–140MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ng KC, Hwang YH, Sheu TWH, Yu CH (2015) Moving particle level-set (MPLS) method for incompressible multiphase flow computation. Comput Phys Commun 196:317–334CrossRefGoogle Scholar
  11. 11.
    Wang L, Jiang Q, Zhang C (2017) Improvement of moving particle semi-implicit method for simulation of progressive water waves. Int J Numer Methods Fluids 85:69–89MathSciNetCrossRefGoogle Scholar
  12. 12.
    Morris JP, Fox PJ, Zhu Y (1997) Modeling low Reynolds number incompressible flows using SPH. J Comput Phys 136:214–226CrossRefzbMATHGoogle Scholar
  13. 13.
    Cummins S, Rudman M (1999) An SPH projection method. J Comput Phys 152:584–607MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Shao S, Lo EYM (2003) Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Adv Water Resour 26:787–800CrossRefGoogle Scholar
  15. 15.
    Liu GR, Liu MB (2003) Smoothed particle hydrodynamics: a meshfree particle method. World Scientific Publishing Co. Pte. Ltd., SingaporeCrossRefzbMATHGoogle Scholar
  16. 16.
    Chiron L, Oger G, de Leffe M, Le Touze D (2018) Analysis and improvements of adaptive particle refinement (APR) through CPU time, accuracy and robustness considerations. J Comput Phys 354:552–575MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sun PN, Colagrossi A, Marrone S, Antuono M, Zhang AM (2018) Multi-resolution delta-plus-SPH with tensile instability control: towards high Reynolds number flows. Comput Phys Commun 224:63–80MathSciNetCrossRefGoogle Scholar
  18. 18.
    Yang X, Liu M, Peng S (2014) Smoothed particle hydrodynamics and element bending group modeling of flexible fibers interacting with viscous fluids. Phys Rev E 90:063011CrossRefGoogle Scholar
  19. 19.
    Tamai T, Koshizuka S (2014) Least squares moving particle semi-implicit method. Comput Part Mech 1:277–305CrossRefGoogle Scholar
  20. 20.
    Fangyuan H, Matsunaga T, Tamai T, Koshizuka S (2017) An ALE particle method using upwind interpolation. Comput Fluids 145:21–36MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zhang S, Morita K, Fukuda K, Shirakawa N (2006) An improved MPS method for numerical simulations of convective heat transfer problems. Int J Numer Methods Fluids 51:31–47CrossRefzbMATHGoogle Scholar
  22. 22.
    Khayyer A, Gotoh H (2009) Modified moving particle semi-implicit methods for the prediction of 2D wave impact pressure. Coast Eng 56:419–440CrossRefGoogle Scholar
  23. 23.
    Lee BH, Park JC, Kim MH, Hwang SC (2011) Step-by-step improvement of MPS method in simulating violent free-surface motions and impact-loads. Comput Methods Appl Mech Eng 200:1113–1125CrossRefzbMATHGoogle Scholar
  24. 24.
    Ng KC, Sheu TWH, Hwang YH (2016) Unstructured moving particle pressure mesh (UMPPM) method for incompressible isothermal and non-isothermal flow computation. Comput Methods Appl Mech Eng 305:703–738MathSciNetCrossRefGoogle Scholar
  25. 25.
    Wei H, Pan W, Rakhsha M, Tian Q, Haiyan H, Negrut D (2017) A consistent multi-resolution smoothed particle hydrodynamics method. Comput Methods Appl Mech Eng 324:279–299MathSciNetGoogle Scholar
  26. 26.
    Obeidat A, Bordas SPA (2018) Three-dimensional remeshed smoothed particle hydrodynamics for the simulation of isotropic turbulence. Int J Numer Methods Fluids 86:1–19MathSciNetCrossRefGoogle Scholar
  27. 27.
    Qingsong T, Liu S (2017) An updated Lagrangian particle hydrodynamics (ULPH) for Newtonian fluids. J Comput Phys 348:493–513MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lian YP, Zhang X, Zhou X, Ma S, Zhao YL (2011) Numerical simulation of explosively driven metal by material point method. Int J Impact Eng 38:238–246CrossRefGoogle Scholar
  29. 29.
    Zhang F, Zhang X, Sze KY, Lian Y, Liu Y (2017) Incompressible material point method for free surface flow. J Comput Phys 330:92–110MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Chen JK, Beraun JE, Carney TC (1999) A corrective smoothed particle method for boundary value problems in heat conduction. Int J Numer Methods Eng 46:231–252CrossRefzbMATHGoogle Scholar
  31. 31.
    Shao JR, Li HQ, Liu GR, Liu MB (2012) An improved SPH method for modeling liquid sloshing dynamics. Comput Struct 100–101:18–26CrossRefGoogle Scholar
  32. 32.
    Zhang ZL, Liu MB (2017) Smoothed particle hydrodynamics with kernel gradient correction for modeling high velocity impact in two- and three-dimensional spaces. Eng Anal Bound Elem 83:141–157MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Liu MB, Li SM (2016) On the modeling of viscous incompressible flows with smoothed particle hydrodynamics. J Hydrodyn 28:731–745CrossRefGoogle Scholar
  34. 34.
    Zhang ZL, Liu MB (2018) A decoupled finite particle method for modeling incompressible flows with free surfaces. Appl Math Model 60:606–633MathSciNetCrossRefGoogle Scholar
  35. 35.
    Chiu PH, Sheu Tony WH (2009) On the development of a dispersion-relation-preserving dual-compact upwind scheme for convection–diffusion equation. J Comput Phys 228:3640–3655CrossRefzbMATHGoogle Scholar
  36. 36.
    Tam CKW, Webb JC (1993) Dispersion-relation-preserving finite difference schemes for computational acoustics. J Comput Phys 107:262–281MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Bhumkar Y, Sheu TWH, Sengupta TK (2014) A dispersion relation preserving optimized upwind compact difference scheme for high accuracy flow simulations. J Comput Phys 278:378–399MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Brambley EJ (2016) Optimized finite-difference (DRP) schemes perform poorly for decaying or growing oscillations. J Comput Phys 324:258–274MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Raithby GD (1976) Skew upstream differencing schemes for problems involving fluid flow. Comput Methods Appl Mech Eng 9:153–164MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Patel MK, Markatos NC, Cross M (1985) Method of reducing false-diffusion errors in convection–diffusion equations. Appl Math Model 9:302–306CrossRefGoogle Scholar
  41. 41.
    Carey C, Scanlon TJ, Fraser SM (1993) SUCCA—an alternative scheme to reduce the effects of multidimensional false diffusion. Appl Math Model 17:263–270CrossRefzbMATHGoogle Scholar
  42. 42.
    Bailey RT (2017) Managing false diffusion during second-order upwind simulations of liquid micromixing. Int J Numer Methods Fluids 83:940–959MathSciNetCrossRefGoogle Scholar
  43. 43.
    Liu KS, Sheu Tony WH, Hwang YH, Ng KC (2017) High-order particle method for solving incompressible Navier–Stokes equations within a mixed Lagrangian–Eulerian framework. Comput Methods Appl Mech Eng 325:77–101MathSciNetCrossRefGoogle Scholar
  44. 44.
    Lancaster P, Salkauskas K (1981) Surfaces generated by moving least squares methods. Math Comput 37:141–158MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Cueto-Felgueroso L, Colominas I, Nogueira X, Navarrina F, Casteleiro M (2007) Finite volume solvers and moving least-squares approximations for the compressible Navier–Stokes equations on unstructured grids. Comput Methods Appl Mech Eng 196:4712–4736MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Chassaing J-C, Khelladi S, Nogueira X (2013) Accuracy assessment of a high-order moving least squares finite volume method for compressible flows. Comput Fluids 71:41–53MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Avesani D, Dumbser M, Bellin A (2014) A new class of moving-least-squares WENO-SPH schemes. J Comput Phys 270:279–299MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Ramirez L, Nogueira X, Khelladi S, Chassaing J-C, Colominas I (2014) A new high-order finite volume method based on moving least squares for the resolution of the incompressible Navier–Stokes equations on unstructured grids. Comput Methods Appl Mech Eng 278:883–901CrossRefzbMATHGoogle Scholar
  49. 49.
    Khayyer A, Gotoh H (2008) Development of CMPS method for accurate water-surface tracking in beaking waves. Coast Eng J 50:179–207CrossRefGoogle Scholar
  50. 50.
    Tanaka M, Masunaga T (2010) Stabilization and smoothing of pressure in MPS method by quasi-compressibility. J Comput Phys 229:4279–4290CrossRefzbMATHGoogle Scholar
  51. 51.
    Khayyer A, Gotoh H (2011) Enhancement of stability and accuracy of the moving particle semi-implicit method. J Comput Phys 230:3093–3118MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Kondo M, Koshizuka S (2011) Improvement of stability in improving particle semi-implicit method. Int J Numer Methods Fluids 65:638–654CrossRefzbMATHGoogle Scholar
  53. 53.
    Monaghan JJ (1994) Simulating free surface flows with SPH. J Comput Phys 110:399–406CrossRefzbMATHGoogle Scholar
  54. 54.
    Lee ES, Moulinec C, Xu R, Violeau D, Laurence D, Stansby P (2008) Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free method. J Comput Phys 227:8417–8436MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Adami S, Hu XY, Adams NA (2013) A transport-velocity formulation for smoothed particle hydrodynamics. J Comput Phys 241:292–307MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Hwang YH (2012) Assessment of diffusion operators in a novel moving particle method. Numer Heat Transf B Fundam 61:329–368Google Scholar
  57. 57.
    Hwang YH (2011) Smoothing difference scheme in a moving particle method. Numer Heat Transf B Fundam 60:203–234CrossRefGoogle Scholar
  58. 58.
    Chu PC, Fan CW (1998) A three-point combined compact difference scheme. J Comput Phys 140:370–399MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Strikwerda JC (1997) High-order-accurate schemes for incompressible viscous flow. Int J Numer Methods Fluids 24:715–734MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Zhuang Y, Sun XH (2001) A high-order fast direct solver for singular poisson equations. J Comput Phys 171:79–94MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Chorin AJ (1968) Numerical solution of the Navier–Stokes equations. Math Comput 22:745–762MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Huang CL, Sheu TWH, Ishikawa T, Yamaguchi T (2011) Development of a particle interaction kernel for convection–diffusion scalar transport equation. Numer Heat Transf B Fundam 60:96–115CrossRefGoogle Scholar
  63. 63.
    Joldes GR, Chowdhury HA, Wittek A, Doyle B, Miller K (2015) Modified moving least squares with polynomial bases for scattered data approximation. Appl Math Comput 266:893–902MathSciNetGoogle Scholar
  64. 64.
    Tyliszczak A (2014) A high-order compact difference algorithm for half-staggered grids for laminar and turbulent incompressible flows. J Comput Phys 276:438–467MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Adami S, Hu XY, Adams NA (2012) A generalized wall boundary condition for smoothed particle hydrodynamics. J Comput Phys 231:7057–7075MathSciNetCrossRefGoogle Scholar
  66. 66.
    Kunz P, Hirschler M, Huber M, Nieken U (2016) Inflow/outflow with Dirichlet boundary conditions for pressure in ISPH. J Comput Phys 326:171–187MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Mitsume N, Yoshimura S, Murotani K, Yamada T (2015) Explicitly represented polygon wall boundary model for the explicit MPS method. Comput Part Mech 2:73–89CrossRefGoogle Scholar
  68. 68.
    Li S, Liu WK (2002) Meshfree and particle methods and their applications. Appl Mech Rev 55:1–34CrossRefGoogle Scholar
  69. 69.
    Liu MB, Liu GR, Lam KY (2003) Constructing smoothing functions in smoothed particle hydrodynamics with applications. J Comput Appl Math 155:263–284MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Liu MB, Liu GR (2006) Restoring particle consistency in smoothed particle hydrodynamics. Appl Numer Math 56:19–36MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle method. Int J Numer Methods Fluids 20:1081–1106MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Ellero M, Serrano M, Espanol P (2007) Incompressible smoothed particle hydrodynamics. J Comput Phys 226:1731–1752MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Quinlan NJ, Lobovsky L, Nestor RM (2014) Development of the meshless finite volume particle method with exact and efficient calculation of interparticle area. Comput Phys Commun 185:1554–1563MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    Armaly BF, Durst F, Pereira JCF, Schonung B (1983) Experimental and theoretical investigation of backward-facing step flow. J Fluid Mech 127:473–496CrossRefGoogle Scholar
  75. 75.
    Ghia U, Ghia KN, Shin CT (1982) High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. J Comput Phys 48:387–411CrossRefzbMATHGoogle Scholar
  76. 76.
    Weller HG, Tabor G, Jasak H, Fureby C (1998) A tensoral approach to computational continuum mechanics using object-oriented techniques. Comput Phys 12:620–631CrossRefGoogle Scholar
  77. 77.
    Ng KC (2009) A collocated finite volume embedding method for simulation of flow past stationary and moving body. Comput Fluids 38:347–357CrossRefzbMATHGoogle Scholar

Copyright information

© OWZ 2018

Authors and Affiliations

  • Rex Kuan-Shuo Liu
    • 1
  • Khai-Ching Ng
    • 2
  • Tony Wen-Hann Sheu
    • 1
    • 3
    • 4
    Email author
  1. 1.Department of Engineering Science and Ocean engineeringNational Taiwan UniversityTaipeiTaiwan
  2. 2.School of EngineeringTaylor’s UniversitySubang JayaMalaysia
  3. 3.Institute of Applied Mathematical SciencesNational Taiwan UniversityTaipeiTaiwan
  4. 4.Center for Advanced Study in Theoretical SciencesNational Taiwan UniversityTaipeiTaiwan

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