# Resolving phase transitions with Discontinuous Galerkin methods

### Submission summary

 As Contributors: Eduardo Grossi · Nicolas Wink Arxiv Link: https://arxiv.org/abs/1903.09503v2 (pdf) Date submitted: 2019-04-24 02:00 Submitted by: Wink, Nicolas Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Computational High-Energy Physics - Theory High-Energy Physics - Phenomenology Approaches: Theoretical, Computational

### Abstract

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:
Has been resubmitted

### Submission & Refereeing History

Resubmission 1903.09503v3 on 4 July 2019

Submission 1903.09503v2 on 24 April 2019

## Reports on this Submission

### Strengths

1. The paper seeks to introduce and discuss the applicability of Discontinuous Galerkin
methods in solving functional renormalization group flow equations. This could in principle be a useful goal.

### Weaknesses

1. Although the basic goal of introducing Discontinuous Galerkin methods to a new audience
could be useful if achieved, it is not clear that the results would be significant enough for
this journal.
2. Unfortunately the introduction to Discontinuous Galerkin methods is too brief to be
understood without already having knowledge of these methods, although it is possible to
fish out some introductory material buried much later in the paper.
2. The paper is applied only to the large-N O(N) models, apart from some brief speculations
at the end of sec. III. Such models are very special - in fact exactly soluble in this context also as the authors themselves review in appendix B. One is left with the impression that these
methods are only applicable to such first order non-linear equations, as appears also to be
true of methods based on a weak solution on which these methods appear to be closely
based. In this case, one is already better off with using the method of appendix B coupled with numerical inversion.
3. The number of typos, mis-spellings, grammatical inaccuracies and other evidences of lack
of care, is so great that it would be impractical to list more than a sample.
4. There are a number of places where the mathematical development and/or physical
reasoning seems also to be suspect.

### Report

The authors' stated intention is "to close the cap [sic: should be "gap"] between the FRG and DG community". As someone from the FRG community, I found sec. IB far too brief to achieve this ambition. Even before this introduction starts there is reference to "conservation
equations" which one gradually realises means something quite specific here (normal use of
the word "conservation" would imply something -like a charge- is held constant) and to a
"weak formulation". These terms are never explained (to be joined by many others e.g. "CFL" conditions and "BDF" methods on p5, NB next sentence: "persevering"->preserving). On the other hand above eqn (23) you actually write "the first derivative of the potential is a
conserved quantity". At face value you seem to be telling me that the object in eqn (23) is a
constant with respect to t? Reference is made to intuition about relative rates of diffusion
versus convection, but the reader is never inducted into this way of thinking i.e. what precisely
this means for the FRG equation and how it relates to physics the reader from the FRG
community may have seen before.

The derivation of eqn (9) is given as integrating by parts eqn (7) to get (8) and then
integrating by parts back again. Since (9) is different from (7) there is clearly more to this, but readers are left to try to figure out why for themselves, or go learn about these methods from
somewhere where it is better explained. Eqn (10) is pulled out of a hat (NB "state the we work" -> state that we work), together with (11) (now in general dimension and not one dimension as earlier? why? - given that the rest of the paper is in one dimension).

Another example: eqn (27) on p5 is not explained, just quoted. Then it turns out it is derived
at the end of appendix C1! This can easily be missed unless one picks up on the one reference
on another matter at the end of sec. IIC on p11. Equation (27) is integrated in eqn (28). It
would appear that the authors assume that $u_L$ and $u_R$ are held constant, although the authors are silent on this point. When is such an assumption justified? (Incidentally in the next
paragraph I am not sure what word you actually meant by "prepositions").

Some more examples: eqn (35) is claimed to be generic for any coupling. Maybe so in large N but not generally. (Next section "It is instructing" -> It is instructive.) Why are Legendre
polynomials used? (App A. NB "Strum-Liouville"-> Sturm-Liouville, here & throughout.)

In many places two different forms of a sentence appear together, such as "are given can
easily be obtained", "convenient [with] respect compared to others" in App A. Many verbs
incorrectly conjugated e.g. "chose"-> choose, "condense" -> condensed, "outline" ->
outlined etc. Wrong words e.g. "underling" -> underlying, "limes" -> limits etc.

The authors need to go through the paper line by line e.g. using an editor that will prompt for all the grammatical inaccuracies. With some substantial improvements, not only to this but
the whole logic and presentation, this paper might find a useful home in a more pedagogically
minded journal. Sifting through this paper in its present form, I saw no evidence to make me
think that the work is suitable for this journal.

### Requested changes

Not applicable

• validity: low
• significance: low
• originality: ok
• clarity: poor
• formatting: good
• grammar: mediocre

### Author:  Nicolas Wink  on 2019-07-03

(in reply to Report 1 on 2019-06-17)
Category:
objection

## General points

Before addressing the individual criticism in the report we would like to outline two points:

1. Despite being depicted as such in the report, the paper is neither meant to be overly pedagogical nor an introduction to the topic. First and foremost, it is a research paper with a slightly longer introduction chapter than usual.
2. The physics aspect of the paper is not mentioned a single time in the report.

## Weaknesses

Although the basic goal of introducing Discontinuous Galerkin methods to a new audience could be useful if achieved, it is not clear that the results would be significant enough for this journal.

This is not intended goal of paper. However, we are aware of the fact that these methods are not well known in the FRG community. Therefore, we included an introduction to the topic. We do think that the research content of the paper is well suited for this journal.

Unfortunately the introduction to Discontinuous Galerkin methods is too brief to be understood without already having knowledge of these methods, although it is possible to fish out some introductory material buried much later in the paper.

The introduction only aims at introducing DG schemes. However, we do agree that the section about numerical fluxes might be a bit short if the reader is not familiar with Finite Volume Methods. Therefore, we have extended this part. We do want to stress however, that we do not aim at giving a broad introduction to the topic. As a direct result, quite some of the technical details are in the Appendix.

The paper is applied only to the large-N O(N) models, apart from some brief speculations at the end of sec. III. Such models are very special - in fact exactly soluble in this context also as the authors themselves review in appendix B. One is left with the impression that these methods are only applicable to such first order non-linear equations, as appears also to be true of methods based on a weak solution on which these methods appear to be closely based. In this case, one is already better off with using the method of appendix B coupled with numerical inversion.

• For the development of numerical methods it is extremely useful and customary to first investigate problems that are either solvable (semi)-analytically or by other means.
• We would like to point out that the method of characteristics is not "better" than the scheme presented here. As a matter of fact, for quite some aspects presented in the paper DG schemes are computationally cheaper, e.g. the resolution of the RG-time evolution.
• The order of the equation and the concept of a weak solution are entirely unrelated.
• DG schemes are by no means restricted to first order differential equations. Since these schemes are almost never blindly applicable to new physical systems it is natural to start at the simplest possible approximation. A prominent use case for these schemes are the compressible Navier–Stokes equations, i.e. a second order non-linear set of equations. Additionally, we have added some references where diffusion equations have been studied with DG schemes.

The number of typos, mis-spellings, grammatical inaccuracies and other evidences of lack of care, is so great that it would be impractical to list more than a sample.

We agree and correct as many as possible.

There are a number of places where the mathematical development and/or physical reasoning seems also to be suspect.

We cannot comment to this general criticism, since the corresponding points are not apparent in the report and we obviously disagree.

# Report

The authors' stated intention is "to close the gap...

This is our intention for the existence of the relevant subsections in the introduction, not the paper.

Even before this introduction starts there is reference to "conservation equations"...

It does not describe something quite specific in this context. The usual, general definition of a conservation law is $\nabla_\mu a^\mu = 0$. It is simply written slight different here. The term conservation law must not be confused with a locally conserved charge.

...These terms are never explained (to be joined by many others e.g. "CFL" conditions and "BDF"...

Not explained at length as we expect the reader to be familiar with the basics of numerical methods for partial differential equations. However, we have extended the existing qualitative explanation of the CFL condition slightly. Written out the acronym for the family of Backward differentiation formula (BDF) methods. We have erased the unnecessary reference to the weak solution before it was introduced.

...At face value you seem to be telling me that the object in eqn (23) is a constant with respect to t?

No, as usual for a conservation law (think of it as e.g. mass conservation for incompressible fluids), it is locally conserved and the rate of change is given by the flux. As pointed out above, while conservation laws and conserved charges are related, they must not be confused with each other.

Reference is made to intuition about relative rates of diffusion versus convection...

We do give some explanations in the introductions (RG flow as convection) and in Section 2.3. We have included an additional cross-reference at the first occurrence and extended the explanation slightly to make the analogy more apparent.

The derivation of eqn (9) is given as integrating by parts eqn (7) to get (8) and then...

While we do not aim at a general introduction to the topic of numerical fluxes, we agree that the explanation given might be to short and have extended the part correspondingly.

now in general dimension and not one dimension as earlier?...

It is still in one dimension, the vector notation is simply a lot more convenient to write the expressions. We have added the explicit expressions for one dimension to avoid confusion.

Another example: eqn (27) on p5 is not explained, just quoted...

We have added the missing cross-reference to the Appendix.

It would appear that the authors assume that $u_L$ and $r_R$ are held constant...

The paper states above (26) that piecewise constant initial conditions define the problem. This assumption holds by construction of the problem. Nevertheless, we added a note that the solution stays trivial piecewise constant for a propagating shock.

eqn (35) is claimed to be generic for any coupling...

We added a disclaimer for Large N theories.

Why are Legendre polynomials used?

Simplest and probably by far the most popular choice for an orthogonal basis in one dimension.

In many places two different forms of a sentence appear together...

We apologize and correct as many errors as possible.