Computational Visual Media

, Volume 4, Issue 3, pp 223–230 | Cite as

Spatially adaptive long-term semi-Lagrangian method for accurate velocity advection

  • Takahiro SatoEmail author
  • Christopher Batty
  • Takeo Igarashi
  • Ryoichi Ando
Open Access
Research Article


We introduce a new advection scheme for fluid animation. Our main contribution is the use of long-term temporal changes in pressure to extend the commonly used semi-Lagrangian scheme further back along the time axis. Our algorithm starts by tracing sample points along a trajectory following the velocity field backwards in time for many steps. During this backtracing process, the pressure gradient along the path is integrated to correct the velocity of the current time step. We show that our method effectively suppresses numerical diffusion, retains small-scale vorticity, and provides better long-term kinetic energy preservation.


fluid simulation advection method of characteristics spatially adaptive integration interpolation error correction 



This work was supported by NSERC (Grant RGPIN-04360-2014) and JSPS KAKENHI (Grant 17H00752). The authors thank Toshiya Hachisuka for insightful discussions.

Supplementary material

41095_2018_117_MOESM1_ESM.mp4 (57.8 mb)
Spatially Adaptive Long-Term Semi-Lagrangian Method for Accurate Velocity Advection


  1. [1]
    Selle, A.; Fedkiw, R.; Kim, B.; Liu, Y.; Rossignac, J. An unconditionally stable MacCormack method. Journal of Scientific Computing Vol. 35, Nos. 2–3, 350–371, 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Stam, J. Stable fluids. In: Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, 121–128, 1999.Google Scholar
  3. [3]
    Xiu, D.; Karniadakis, G. E. A semi-Lagrangian highorder method for Navier–Stokes equations. Journal of Computational Physics Vol. 172, No. 2, 658–684, 2001.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Tessendorf, J.; Pelfrey, B. The characteristic map for fast and efficient VFX fluid simulations. In: Proceedings of the Computer Graphics International Workshop on VFX, Computer Animation, and Stereo Movies, 2011.Google Scholar
  5. [5]
    Bridson, R. Fluid Simulation for Computer Graphics, 2nd edn. Taylor & Francis, 2015.Google Scholar
  6. [6]
    Shu, C. W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics, Vol. 1697. Quarteroni, A. Ed. Springer Berlin Heidelberg, 325–432, 1998.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Takewaki, H.; Yabe, T. The cubic-interpolated pseudo particle (CIP) method: Application to nonlinear and multi-dimensional hyperbolic equations. Journal of Computational Physics Vol. 70, No. 2, 355–372, 1987.CrossRefzbMATHGoogle Scholar
  8. [8]
    Foster, N.; Fedkiw, R. Practical animation of liquids. In: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, 23–30, 2001.Google Scholar
  9. [9]
    Heo, N.; Ko, H.-S. Detail-preserving fully-Eulerian interface tracking framework. In: Proceedings of the ACM SIGGRAPH Asia 2010 Papers, Article No. 176, 2010.Google Scholar
  10. [10]
    Hachisuka, T. Combined Lagrangian–Eulerian approach for accurate advection. In: Proceedings of the ACM SIGGRAPH 2005 Posters, Article No. 114, 2005.CrossRefGoogle Scholar
  11. [11]
    Hachisuka, T. Advection equation solver using mapping functions. Thesis for Bachelor of Engineering. 2006. Available at Scholar
  12. [12]
    Zhu, Y.; Bridson, R. Animating sand as a fluid. In: Proceedings of the ACM SIGGRAPH 2005 Papers, 965–972, 2005.CrossRefGoogle Scholar
  13. [13]
    Mullen, P.; Crane, K.; Pavlov, D.; Tong, Y.; Desbrun, M. Energy-preserving integrators for fluid animation. In: Proceedings of the ACM SIGGRAPH 2009 Papers, Article No. 38, 2009.Google Scholar
  14. [14]
    Zhang, X.; Bridson, R.; Greif, C. Restoring the missing vorticity in advection-projection fluid solvers. ACM Transactions on Graphics Vol. 34, No. 4, Article No. 52, 2015.Google Scholar
  15. [15]
    Lentine, M.; Zheng, W.; Fedkiw, R. A novel algorithm for incompressible flow using only a coarse grid projection. In: Proceedings of the ACM SIGGRAPH 2010 Papers, Article No. 114, 2010.Google Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Takahiro Sato
    • 1
    Email author
  • Christopher Batty
    • 2
  • Takeo Igarashi
    • 1
  • Ryoichi Ando
    • 3
  1. 1.The University of TokyoTokyoJapan
  2. 2.University of WaterlooWaterlooCanada
  3. 3.National Institute of InformaticsTokyoJapan

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