## SciPost Submission Page

# Resolving phase transitions with Discontinuous Galerkin methods

### by Eduardo Grossi, Nicolas Wink

#### This is not the current version.

### Submission summary

As Contributors: | Eduardo Grossi · Nicolas Wink |

Arxiv Link: | https://arxiv.org/abs/1903.09503v2 |

Date submitted: | 2019-04-24 |

Submitted by: | Wink, Nicolas |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | High-Energy Physics - Theory |

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:

### Submission & Refereeing History

## Reports on this Submission

### Anonymous Report 1 on 2019-6-17 Invited Report

### 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

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

## General points

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

## Weaknesses

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.

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.

We agree and correct as many as possible.

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

## Report

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

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.

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.

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.

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.

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.

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.

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

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.

We added a disclaimer for Large N theories.

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

We apologize and correct as many errors as possible.