Resolving phase transitions with Discontinuous Galerkin methods

Weaknesses

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.

We have checked this in the given setting of the second order phase transition, the convergence properties are independent of the final RG-time. However, a relative comparison between different RG-times would mostly check the convergence properties of the external library we used for solving the resulting set of odes, as stated in the paper. We don't think it is worth repeating the study for a solution with a shock, because obtaining a reference solution that is suitable for comparison comes with considerable amount of work. After all, the actual solution is also obtained numerically when using the method of characteristics. We added a statement clarifying this situation in the text.

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

We corrected as many typos and grammatical inaccuracies as possible.

Report

As a minor suggestion, it might be useful to use $\tau$ instead of $t$ for (minus) the RG time to avoid confusion.

We would prefer to keep the current notation.

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].

The suggested paper is already cited, categorized as vertex expansion. Most of the works that are considered as vertex expansion also resolve the field dependence to at least some extent. We have changed the wording slightly to reflect better how these terms are understood.

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.

We added a sentence explaining this.

Before eq. 10, a reference for the "Local Lax-Friedrichs flux" might be useful.

We added the original reference.

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.

We extended this part appropriately.

We extended this part appropriately.

Section IIA, second sentence: the action is expanded in powers of gradients of the field, not the field in terms of gradients.

Fixed the typo.

The tensor structure of the regulator should be specified.

We clarified that we used a diagonal regulator.

Before eq. 20, I suspect that $(t,\rho)$ should read $V(t,\rho)$.

Fixed the typo.

CFL should be spelled out once.

We included the full name once.

Fig. 5 is discussed after Fig. 6, they should thus be reordered.

This was done for styling reasons on purpose.

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.

We fully agree that it makes no sense to have this comparison at t=1.75. We updated the figure and the coefficient table to the comparison at $t=4$. The coefficients still agree within error tolerances, except for $a_1$. This is expected at different RG-times, since it simply accounts for the accumulated error from the RG-time integration.

Fig. 8c) and d): there seem to be numerical artefacts near vanishing field - these should be explained somewhere.

This is a result of the WENO limiter we used in order to remove the Gibbs oscillations from the plot. We have added a sentence clarifying this.

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].

We added references, in particular also [Phys.Rev. D90 (2014) no.7, 076002], which seems to be the first work considering this.

The authors use Legendre polynomials. While these provide...

For technical convenience since it only effects the subleading part and does not change any fundamental properties of the scheme. Additionally, they are the standard choice in Finite Element based approaches.

Eqs. C1 and C2: it seems that coming from the first to the second equation, ...

The integrands in C2 are continuous by design and an additional term arises from the discontinuity that's present, which can be seen in C3. We added a sentence clarifying that the interval needs to chosen sufficiently small such that only one discontinuity is present.

Eq. C12: it wasn't clear to me what $\lambda_k$ refers to in this equation.

We clarified this, it's the eigenvalue of the Sturm-Liouville equation, introduced below A1.

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.

We corrected as many typos and grammatical inaccuracies as possible.

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dc_galerkin_frg.pdf