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Resolving phase transitions with Discontinuous Galerkin methods

by Eduardo Grossi, Nicolas Wink

Submission summary

As Contributors: Eduardo Grossi · Nicolas Wink
Arxiv Link: (pdf)
Date submitted: 2019-07-04 02:00
Submitted by: Wink, Nicolas
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Computational
  • High-Energy Physics - Theory
  • High-Energy Physics - Phenomenology
Approaches: Theoretical, Computational


We demonstrate the applicability and advantages of Discontinuous Galerkin (DG) schemes in the context of the Functional Renormalization Group (FRG). We investigate the $O(N)$-model in the large $N$ limit. It is shown that the flow equation for the effective potential can be cast into a conservative form. We discuss results for the Riemann problem, as well as initial conditions leading to a first and second order phase transition. In particular, we unravel the mechanism underlying first order phase transitions, based on the formation of a shock in the derivative of the effective potential.

Current status:
Editor-in-charge assigned

List of changes

- Extended explanation of numerical flux
- Slight extension of several other explanations
- Fixed typos

Submission & Refereeing History

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Resubmission 1903.09503v3 on 4 July 2019

Reports on this Submission

Anonymous Report 1 on 2019-8-1 (Invited Report)


1) The paper introduces discontinuous Galerkin methods into the context of the FRG. This will prove very useful since we are often dealing with equations which develop discontinuities in some derivative, and a good numerical treatment is key for both qualitatively and quantitatively correct results.

2) Results on the O(N) model in the large N limit can be benchmarked against the analytic solution, which illustrates the stability of the numerical methods.

3) The paper studies both first and second order phase transitions, which spans virtually all cases of phase transitions one encounters. Interesting results on shock formation and rarefaction waves are presented.


1) It would have been interesting to push the numerics to larger values of the RG time to check how long the numerical error is under control, especially in the case where shocks form.

2) As the first referee also said, the language level is mediocre, there are still a lot of typos/grammatical inaccuracies etc.


Overall, I disagree with referee 1 - I find the introduction of the DG methods into the FRG context performed in the paper worthwhile and successful. The O(N) model in the large N limit is a well-known and analytically solvable testing grounds where numerical techniques can be tested in a controlled way. Of particular interest is the study of the formation of shocks and rarefaction waves, which is, to my knowledge, new in the FRG context. The paper and the methods it introduces will thus certainly make an impact on future research when it comes to studying phase transitions that feature non-smooth behaviour.

Having said that, there are some points that I would like the authors to address before I can recommend that the paper can be published.

- As a minor suggestion, it might be useful to use $\tau$ instead of $t$ for (minus) the RG time to avoid confusion.
- In section IA, where the authors discuss different approximation schemes, it seems fair to me to refer to the BMW scheme which resolves both momentum and field dependence, see e.g. [Phys.Lett. B632 (2006) 571-578].
- It might help the inexperienced reader to point out more explicitly what the Galerkin part (eq. 7) and the discontinuous part of the scheme is.
- Before eq. 10, a reference for the "Local Lax-Friedrichs flux" might be useful.
- In section II, for the examples it should be clarified which dimensions they correspond to (Standard Model: d=4, condensed matter: typically d=3) and some references might be added.
- Section IIA, second sentence: the action is expanded in powers of gradients of the field, not the field in terms of gradients.
- The tensor structure of the regulator should be specified.
- Before eq. 20, I suspect that $(t,\rho)$ should read $V(t,\rho)$.
- CFL should be spelled out once.
- Fig. 5 is discussed after Fig. 6, they should thus be reordered.
- Fig. 5 shows the convergence of the numerical scheme for different numbers of elements and orders. In the text it says that this is taken at $t=1.75$, at the onset of the flattening of the potential. In this regime it is clear that the accuracy is very high since the potential is still smooth enough, and standard pseudo-spectral methods should work equally well up to this point. It would be more honest to report on the numerical error at the end point of the flow at $t=4$. (*)
- Fig. 8c) and d): there seem to be numerical artefacts near vanishing field - these should be explained somewhere. (*)
- Right before section IV: the authors mention the impact of fermions e.g. by a field dependent Yukawa coupling. Some references here seem to be adequate, e.g. [Phys.Rev. D94 (2016) no.3, 034016, Phys. Rev. D 91, 125003 (2015), Phys. Rev. B 94, 245102 (2016), Eur.Phys.J. C77 (2017) no.11, 743].
- The authors use Legendre polynomials. While these provide exponential convergence for smooth functions, the subleading part of the rate of convergence is worse than that of Chebyshev polynomials, see [Boyd: Chebyshev and Fourier Spectral Methods, Chapter 2.13]. Why do the authors nevertheless use Legendre polynomials?
- Eqs. C1 and C2: it seems that coming from the first to the second equation, the authors have employed the fundamental theorem of calculus. This is however only valid for continuous integrands. In general, the flux might contain a discontinuity, inherited from its dependence on u. This would give rise to additional terms in eq. C2. Can the authors elaborate on why these terms generically don't appear? (*)
- Eq. C12: it wasn't clear to me what $\lambda_k$ refers to in this equation.
- Last but not least, the manuscript still contains an awful lot of spelling mistakes and grammatical errors. I can only recommend (as referee 1) to carefully read and improve the language of the paper (main text and appendices) to improve its readability.

Requested changes

See report. The most important points are marked with a (*).

  • validity: high
  • significance: high
  • originality: good
  • clarity: good
  • formatting: excellent
  • grammar: below threshold

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