Extension of Smoothed Particle Hydrodynamics (SPH), Mathematical Background of Vortex Blob Method (VBM) and Moving Particle Semi-Implicit (MPS)

ABSTRACT

SPH has a reasonable mathematical background. Although VBM and MPS are similar to SPH, their ma-thematical backgrounds seem fragile. VBM has some problems in treating the viscous diffusion of vortices but is known as a practical method for calculating viscous flows. The mathematical background of MPS is also not sufficient. Not with standing, the numerical results seem reasonable in many cases. The problem common in both VBM and MPS is that the space derivatives necessary for calculating viscous diffusion are not estimated reasonably, although the treatment of advection is mathematically correct. This paper discusses a method to estimate the above mentioned problem of how to treat the space derivatives. The numerical results show the comparison among FDM (Finite Difference Method), SPH and MPS in detail. In some cases, there are big differences among them. An extension of SPH is also given.

SPH has a reasonable mathematical background. Although VBM and MPS are similar to SPH, their ma-thematical backgrounds seem fragile. VBM has some problems in treating the viscous diffusion of vortices but is known as a practical method for calculating viscous flows. The mathematical background of MPS is also not sufficient. Not with standing, the numerical results seem reasonable in many cases. The problem common in both VBM and MPS is that the space derivatives necessary for calculating viscous diffusion are not estimated reasonably, although the treatment of advection is mathematically correct. This paper discusses a method to estimate the above mentioned problem of how to treat the space derivatives. The numerical results show the comparison among FDM (Finite Difference Method), SPH and MPS in detail. In some cases, there are big differences among them. An extension of SPH is also given.

Cite this paper

Isshiki, H. (2014) Extension of Smoothed Particle Hydrodynamics (SPH), Mathematical Background of Vortex Blob Method (VBM) and Moving Particle Semi-Implicit (MPS).*American Journal of Computational Mathematics*, **4**, 414-445. doi: 10.4236/ajcm.2014.45036.

Isshiki, H. (2014) Extension of Smoothed Particle Hydrodynamics (SPH), Mathematical Background of Vortex Blob Method (VBM) and Moving Particle Semi-Implicit (MPS).

References

[1] Isshiki, H. (2011) Discrete Differential Operators on Irregular Nodes (DDIN). International Journal for Numerical Methods in Engineering, 88, 1323-1343. http://dx.doi.org/10.1002/nme.3225

[2] Gingold, R.A. and Monaghan, J.J. (1977) Smoothed Particle Hydrodynamics: Theory and Application to Non-Spherical Stars. Monthly Notices of the Royal Astronomical Society, 181, 375-389.

http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1977MNRAS.181..375G&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf

http://dx.doi.org/10.1093/mnras/181.3.375

[3] Lucy, L.B. (1977) A Numerical Approach to the Testing of the Fission Hypothesis. The Astronomical Journal, 82, 1013-1024.

http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1977AJ.....82.1013L&defaultprint=YES&filetype=.pdf

http://dx.doi.org/10.1086/112164

[4] Chorin, A.J. (1973) Numerical Study of Slightly Viscous Flow. Journal of Fluid Mechanics, 57, 785-796.

http://dx.doi.org/10.1017/S0022112073002016

[5] Yokoyama, M., Kubota, Y., Kikuchi, K., Yagawa, G. and Mochizuki, O. (2014) Some Remarks on Surface Conditions of Solid Body Plunging into Water with Particle Method. Advanced Modeling and Simulation in Engineering Sciences, 1, 2-14. http://www.amses-journal.com/content/pdf/2213-7467-1-9.pdf

[6] Ng, K.C., Hwang, Y.H. and Sheu, T.W.H. (2014) On the Accuracy Assessment of Laplacian Models in MPS. Computer Physics Communications, 185, 2412-2426.

[7] Isshiki, H., Nagata, S. and Imai, Y. (2014) Solution of Viscous Flow around a Circular Cylinder by a New Integral Representation Method (NIRM). The Association for Japan Exchange and Teaching, 2, 60-82.

file:///C:/Users/l/Downloads/983-5001-1-PB%20(2).pdf

[8] Koshizuka, S. and Oka, Y. (1996) Moving Particle Semi-Implicit Method for Fragmentation of Incompressible Fluid. Nuclear Science and Engineering, 123, 421-434.

[9] Moscardini, L. and Dolag, K. (2011) Chapter 4: Cosmology with Numerical Simulations. In: Matarrese, S., Colpi, M., Gorini, V. and Moschella, U., Eds., Dark Matter and Dark Energy, Springer, Berlin, 217-237.

[10] Monaghan, J.J. (1992) Smoothed Particle Hydrodynamics. Annual Review of Astronomy and Astrophysics, 30, 543-573.

http://dx.doi.org/10.1146/annurev.aa.30.090192.002551

[11] Wikipedia, Burgers’ Equation. http://en.wikipedia.org/wiki/Burgers'_equation

[1] Isshiki, H. (2011) Discrete Differential Operators on Irregular Nodes (DDIN). International Journal for Numerical Methods in Engineering, 88, 1323-1343. http://dx.doi.org/10.1002/nme.3225

[2] Gingold, R.A. and Monaghan, J.J. (1977) Smoothed Particle Hydrodynamics: Theory and Application to Non-Spherical Stars. Monthly Notices of the Royal Astronomical Society, 181, 375-389.

http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1977MNRAS.181..375G&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf

http://dx.doi.org/10.1093/mnras/181.3.375

[3] Lucy, L.B. (1977) A Numerical Approach to the Testing of the Fission Hypothesis. The Astronomical Journal, 82, 1013-1024.

http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1977AJ.....82.1013L&defaultprint=YES&filetype=.pdf

http://dx.doi.org/10.1086/112164

[4] Chorin, A.J. (1973) Numerical Study of Slightly Viscous Flow. Journal of Fluid Mechanics, 57, 785-796.

http://dx.doi.org/10.1017/S0022112073002016

[5] Yokoyama, M., Kubota, Y., Kikuchi, K., Yagawa, G. and Mochizuki, O. (2014) Some Remarks on Surface Conditions of Solid Body Plunging into Water with Particle Method. Advanced Modeling and Simulation in Engineering Sciences, 1, 2-14. http://www.amses-journal.com/content/pdf/2213-7467-1-9.pdf

[6] Ng, K.C., Hwang, Y.H. and Sheu, T.W.H. (2014) On the Accuracy Assessment of Laplacian Models in MPS. Computer Physics Communications, 185, 2412-2426.

[7] Isshiki, H., Nagata, S. and Imai, Y. (2014) Solution of Viscous Flow around a Circular Cylinder by a New Integral Representation Method (NIRM). The Association for Japan Exchange and Teaching, 2, 60-82.

file:///C:/Users/l/Downloads/983-5001-1-PB%20(2).pdf

[8] Koshizuka, S. and Oka, Y. (1996) Moving Particle Semi-Implicit Method for Fragmentation of Incompressible Fluid. Nuclear Science and Engineering, 123, 421-434.

[9] Moscardini, L. and Dolag, K. (2011) Chapter 4: Cosmology with Numerical Simulations. In: Matarrese, S., Colpi, M., Gorini, V. and Moschella, U., Eds., Dark Matter and Dark Energy, Springer, Berlin, 217-237.

[10] Monaghan, J.J. (1992) Smoothed Particle Hydrodynamics. Annual Review of Astronomy and Astrophysics, 30, 543-573.

http://dx.doi.org/10.1146/annurev.aa.30.090192.002551

[11] Wikipedia, Burgers’ Equation. http://en.wikipedia.org/wiki/Burgers'_equation