Resolving phase transitions with Discontinuous Galerkin methods

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.

Bosonic fluctuations in the $( 1 + 1 )$-dimensional Gross-Neveu(-Yukawa) model at varying $μ$ and $T$ and finite $N$

“…Numerical computations with the LPA flow equation within this work are based on a reformulation of the corresponding FRG flow equation as a conservation law put forward by some authors and collaborators in Refs. [110,[142][143][144][145]154]. The flow equation in this setup manifests as a non-linear diffusion equation with a sink/source term, where the diffusive contributions can be clearly attributed to bosonic quantum fluctuations, while fermionic fluctuations enter the flow via a sink/source term.…”

“…( 15). However, only recently it was found by some of the authors and their collaborators [110,[142][143][144][145]154] that the RG flow equation for the scale dependent effective potential for a large class of models from QFT can be recast as a conservation law in terms of a (advection-)diffusion-source/sink equation and be entirely understood in terms of fluid dynamic notions 15 . A fluid dynamic reinterpretation is as advantageous for several reasons.…”

“…Refs. [110,142,144,145,154], the huge gradients introduced via the chemical potential induce a shock wave that propagates towards smaller |σ|.…”

“…The KT scheme is an established numerical scheme, that can cope with the aforementioned challenges. Furthermore, the KT scheme will allow for a straightforward generalization of our numerics, if one extends the GNY model to a model with continuous chiral symmetry group and related Goldstone modes, which enter the LPA RG flow equation as non-linear advective contributions [110,142,144,145,154]. The original KT scheme, which was presented by A. Kurganov and E. Tadmor in Ref.…”

“…Of course, we do not have any exact reference values at finite N , which can be used as benchmark results to precisely quantify relative (numerical) errors for the GNY model, similar to what is done in Refs. [110,[142][143][144][145]155]. Still, we can take reasonable IR observables, like the square of the σ-curvature mass m 2 σ = ∂ σ u(t, σ) σmin at the IR minimum σ min , and study their (in-)dependence of the (numerical) parameters.…”