Comparison of different iterative schemes for ISPH based on Rankine source solution

  • Zheng, Xing (College of Shipbuilding Engineering, Harbin Engineering University) ;
  • Ma, Qing-wei (College of Shipbuilding Engineering, Harbin Engineering University) ;
  • Duan, Wen-yang (College of Shipbuilding Engineering, Harbin Engineering University)
  • Received : 2016.06.03
  • Accepted : 2016.10.23
  • Published : 2017.07.31


Smoothed Particle Hydrodynamics (SPH) method has a good adaptability for the simulation of free surface flow problems. There are two forms of SPH. One is weak compressible SPH and the other one is incompressible SPH (ISPH). Compared with the former one, ISPH method performs better in many cases. ISPH based on Rankine source solution can perform better than traditional ISPH, as it can use larger stepping length by avoiding the second order derivative in pressure Poisson equation. However, ISPH_R method needs to solve the sparse linear matrix for pressure Poisson equation, which is one of the most expensive parts during one time stepping calculation. Iterative methods are normally used for solving Poisson equation with large particle numbers. However, there are many iterative methods available and the question for using which one is still open. In this paper, three iterative methods, CGS, Bi-CGstab and GMRES are compared, which are suitable and typical for large unsymmetrical sparse matrix solutions. According to the numerical tests on different cases, still water test, dam breaking, violent tank sloshing, solitary wave slamming, the GMRES method is more efficient than CGS and Bi-CGstab for ISPH method.


Supported by : National Natural Science Funds of China


  1. Bonet, J., Lok, T.S., 1999. Variational and momentum preservation aspects of smooth particle hydrodynamics formulation. Comput. Methods Appl. Mech. Eng. 180, 97-115.
  2. Colagrossi, A., Landrini, M., 2003. Numerical simulation of interfacial flows by smoothed particle hydrodynamics. J. Comput. Phys. 191, 448-475.
  3. Cummins, S.J., Rudman, M., 1999. An SPH projection method. J. Comput. Phys. 152, 584-607.
  4. Ellero, M., Serrano, M., Espanol, P., 2007. Incompressible smoothed particle hydrodynamics. J. Comput. Phys. 226, 1731-1752.
  5. Ferrand, M., Laurence, D.R., Rogers, B.D., Violeau, D., Kassiotis, C., 2013. Unified semi-analytical wall boundary conditions for inviscid, laminar or turbulent flows in the meshless SPH method. Int. J. Numer. Meth. Fluids 71, 446-472.
  6. Fujino, S., 2002. GPBICG (m, l): a hybrid of BiCGSTAB and GPBiCG methods with efficiency and robustness. Appl. Numer. Math. 41, 107-117.
  7. Hori, C., Gotoh, H., Ikari, H., Khayyer, A., 2011. GPU-acceleration for moving particle semi-implicit method. Comput. Fluids 51, 174-183.
  8. Hu, X.Y., Adams, N.A., 2007. An incompressible multi-phase SPH method. J. Comput. Phys. 227 (2), 264-278.
  9. Khayyer, A., Gotoh, H., 2011. Enhancement of stability and accuracy of the moving particle semi-implicit method. J. Comput. Phys. 230, 3093-3118.
  10. Khayyer, A.,Gotoh,H., Shao, S.D., 2008. Corrected incompressible SPHmethod for accurate water-surface tracking in breaking waves. Coast. Eng. 55 (3), 26-50.
  11. Kishev, Z.R., Hu, C.H., Kashiwagi, M., 2006. Numerical simulation of violent sloshing by a CIP-based method. J. Mar. Sci. Technol. 11, 111-122.
  12. Koshizuka, S., Oka, Y., 1996. Moving particle semi-implicit method for fragmentation of incompressible fluid. Nucl. Sci. Eng. 123 (3), 421-434.
  13. Koshizuka, S., Ikeda, H., Oka, Y., 1999. Numerical analysis of fragmentation mechanisms in vapour explosions. Nucl. Eng. Des. 189 (1-3), 423-433.
  14. Lee, E.S., Moulinec, C., Xu, R., Violeau, D., Laurence, D., Stansby, P., 2008. Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free particle method. J. Comput. Phys. 227, 8417-8436.
  15. Liu, X., Xu, H.H., Shao, S.D., Lin, P.Z., 2013. An improved incompressible SPH model for simulation of wave - structure interaction. Comput. Fluids 71, 113-123.
  16. Lind, S.J., Xu, R., Stansby, P.K., Rogers, B.D., 2012. Incompressible smoothed particle hydrodynamics for free-surface flows: a generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves. J. Comput. Phys. 231, 1499-1523.
  17. Liu, M.B., Liu, G.R., 2006. Restoring particle consistency in smoothed particle hydrodynamics. Appl. Numer. Math. 56, 19-36.
  18. Ma, Q.W., Zhou, J., 2009. MLPG_R method for numerical simulation of 2D breaking waves. Comput. Model. Eng. Sci. 43 (3), 277-304.
  19. Martin, J.C., Moyce,W.J., 1952. Part IV: an experimental study of the collapse of liquid columns on a rigid horizontal plane. Phil. Trans. R. Soc. Lond. A 244 (882), 312-324.
  20. Mittal, R.C., Alaurdi, A.H., 2003. An efficient method for constructing an ILU preconditioner for solving large sparse nonsymmetric linear systems by the GMRES method. Comput. Math. Appl. 45, 1757-1772.
  21. Monaghan, J.J., 1994. Simulation free surface flows with SPH. J. Compu. Phys. 110 (4), 399-406.
  22. Oger, G., Doring, M., Alessandrini, B., Ferrant, P., 2007. An improved SPH method: towards higher order convergence. J. Comput. Phys. 225, 1472-1492.
  23. Rafiee, A.R., Cummins, S., Rudman, M., Thiagarajan, K., 2012. Comparative study on the accuracy and stability of SPH schemes in simulating energetic free-surface flows. Eur. J. Mech. B/Fluids 36, 1-16.
  24. Saad, Y., 2003. Iterative Methods for Sparse Linear Systems (Second Version). Society for Industrial and Applied Mathematic.
  25. Saad, Y., Schultz, M.H., 1986. GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856-869.
  26. Schwaiger, H.F., 2008. An implicit corrected SPH formulation for thermal diffusion with linear free surface boundary conditions. Int. J. Numer. Meth. Engng. 75, 647-671.
  27. Shadloo, M.S., Zainali, A., Yildiz, M., Suleman, A., 2012. A robust weakly compressible SPH method and its comparison with an incompressible SPH. Int. J. Numer. Methods Eng. 89 (8), 939-956.
  28. Shao, S.D., 2009. Incompressible SPH simulation of water entry of a free-falling object. Int. J. Numer. Meth. Fluids 59 (1), 91-115.
  29. Shao, S.D., Lo Edmond, Y.M., 2003. Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Adv. Water Resour. 26 (7), 787-800.
  30. Shao, S.D., Ji, C.M., Graham, D.I., Reeve, D.E., James, P.W., Chadwick, A.J., 2006. Simulation of wave overtopping by an incompressible SPH model. Coast. Eng. 53 (9), 723-735.
  31. Sleijpen, G.L., Fokkema, D.R., 1993. BiCGstab(L) for linear equations involving unsymmetric matrices with complex spectrum. Electron. Trans. Numer. Analysis 1, 11-32.
  32. Sonneveld, P., 1989. CGS, A fast Lanczos-type solver for nonsymetric linear systems. SIAM J. Sci. Stat. Comput. 10, 36-52.
  33. Sonneveld, P., Van Gijzen, M.B., 2009. IDR(s): a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations. SIAM J. Sci. Comput. 31 (2), 1035-1062.
  34. Spyropoulos, A.N., Palyvos, J.A., Boudouvis, A.G., 2004. Bifurcation detection with the (un) preconditioned GMRES(m). Comput. Methods Appl. Mech. Eng. 193, 4707-4716.
  35. Vogel, J.A., 2007. Flexible BiCG and flexible Bi-CGSTAB for nonsymmetric linear systems. Appl. Math. Comput. 188, 226-233.
  36. Van der Vorst, H.A., 1992. Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for solution of non-symmetric linear system. SIAM J. Sci. Stat. Comput. 13, 631-644.
  37. Xu, R., Stansby, P., Laurence, D., 2009. Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach. J. Comput. Phys. 228 (18), 6703-6725.
  38. Zhang, S., Morita, K., Kenji, F., Shirakawa, N., 2006. An improved mps method for numerical simulations of convective heat transfer problems. Int. J. Numer. Methods Fluids 51, 31-47.
  39. Zheng, X., Ma, Q.W., Duan, W.Y., 2014. Incompressible SPH method based on Rankine source solution for violent water wave simulation. J. Comput. Phys. 276, 291-314.
  40. Zheng, X., Hu, Z.H., Ma, Q.W., Duan, W.Y., 2015. Incompressible SPH based on Rankine source solution for water wave impact simulation. Procedia Eng. 126, 650-654.