what other recent sources of literature on this topic would you recommend?

This is where I'm starting from:

Leveque's article: HRIC scheme

But the related articles seem to be a bit dated (some up to 20 years). Can anyone with experience with such higher order schemes suggest a direction in which I could search?

  • $\begingroup$ Have you tried using a Volume of Fluid Scheme? $\endgroup$
    – Paul
    Mar 16, 2012 at 1:56
  • $\begingroup$ VoF is geometrical.. I'm asking about numerical approaches to advecting the marker function.. VoF volume fraction becomes something different than a volume fraction as soon as it is not bounded between 0 1 and constricted to a single cell layer, which happens for all available highres. or flux limiting numerical advection schemes... still, there is a wide range of aplication for those schemes so I'm kind of interested ... $\endgroup$
    – tmaric
    Mar 28, 2012 at 11:18

1 Answer 1


I take it from your interest in the article that you are interested in incompressible two-phase flows which has quite a bit of literature on it. One of the more common approaches and arguably most straight forward is to employ a level-set method. I am not sure what the best paper on this is currently to suggest unfortunately. Also, if you are interested in allowing the two fluids to mix you may try searching for volume-of-fluid approaches. If you are actually interested in compressible two-phase flow (such as the Euler gas dynamics equations) then I can suggest some relatively recent papers on the subject:

  1. Saurel, R., Franquet, E., Daniel, E. & Metayer, O. L. A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations. Journal of Computational Physics 223, 822–845 (2007).
  2. Schwendeman, D., Wahle, C. & Kapila, A. The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow. Journal of Computational Physics 212, 490–526 (2006).
  3. Abgrall, R. & Karni, S. Computations of Compressible Multifluids. Journal of Computational Physics 169, 594–623 (2001).
  4. Shyue, K.-M. A fluid-mixture type algorithm for barotropic two-fluid flow problems. Journal of Computational Physics 200, 718–748 (2004).
  • $\begingroup$ Thanks, I'm familiar with level set, I'm implementing vof on unstructured mesh and I'm also interested in front tracking. This is however a higher order scheme development for advecting scalar fields with sharp gradients which is of 2nd order of accuracy for smooth fields... can it be that this is a literature dead end? I mean, if the field Leveque is advecting is a density distribution for a two phase system, it is an interesting alternative... $\endgroup$
    – tmaric
    Mar 15, 2012 at 22:01
  • $\begingroup$ There are also DG methods and WENO finite volume methods that can reach higher orders than 2. Are you interested mostly in methods for the advection portion of the fluid flow? $\endgroup$ Mar 16, 2012 at 2:38
  • $\begingroup$ DG methods are great.. I've seen a presentation about a DG based level set method that requires no re-initialization.. I'm interested in the DNS of two phase flows in general... but from the side of the numerical differencing approach to advection, it would be interesting to try out some of those higher order schemes for advection. $\endgroup$
    – tmaric
    Mar 20, 2012 at 9:15
  • $\begingroup$ I've found something along these lines: Flux Corrected Transport (FCT) based VoF method. I'll accept your answer, thanks for the effort! :) $\endgroup$
    – tmaric
    Mar 20, 2012 at 9:24

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