## Renormalization group methods for the analysis of PDEs

### May 3, 2007

Here are a few papers that discuss the renormalization group asymptotic analysis of PDEs in general, and phase field methods in particular:

- Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory. L-Y Chen, Nigel Goldenfeld, and Y Oono. Phys Rev E, 54, 1, 376, July 1996.

Perturbative renormalization group theory is developed as a unified tool for global asymptotic analysis. With numerous examples, we illustrate its application to ordinary differential equation problems involving multiple scales, boundary layers with technically difficult asymptotic matching, and WKB analysis. In contrast to conventional methods, the renormalization group approach requires neither ad hoc assumptions about the structure of perturbation series nor the use of asymptotic matching. Our renormalization group approach provides approximate solutions which are practically superior to those obtained conventionally, although the latter can be reproduced, if desired, by appropriate expansion of the renormalization group approximant. We show that the renormalization group equation may be interpreted as an amplitude equation, and from this point of view develop reductive perturbation theory for partial differential equations describing spatially extended systems near bifurcation points, deriving both amplitude equations and the center manifold.

- Renormalization group approach to multiscale simulation of polycrystalline materials using the phase field crystal model. Nigel Goldenfeld, Badrinarayan P Athreya, and Jonathan A Dantzig. Phys Rev E, 72, 020601(R), 2005.

We propose a computationally efficient approach to multiscale simulation of polycrystalline materials, based on the phase field crystal model. The order parameter describing the density profile at the nanoscale is reconstructed from its slowly varying amplitude and phase, which satisfy rotationally covariant equations derivable from the renormalization group. We validate the approach using the example of two-dimensional grain nucleation and growth.

- Renormalization-group theory for the phase-field crystal equation. Badrinarayan P Athreya, Nigel Goldenfeld, and Jonathan A Dantzig. Phys Rev E, 74, 011601, 2006.

We derive a set of rotationally covariant amplitude equations for use in multiscale simulation of the two-dimensional phase-field crystal model by a variety of renormalization-group (RG) methods. We show that the presence of a conservation law introduces an ambiguity in operator ordering in the RG procedure, which we show how to resolve. We compare our analysis with standard multiple-scale techniques, where identical results can be obtained with greater labor, by going to sixth order in perturbation theory, and by assuming the correct scaling of space and time.

- Reductive use of renormalization group. K Nozaki, Y Oono, and Y Shiwa. Phys Rev E, 62, R4501, 2000.

It has been recognized that singular perturbation and reductive perturbation can be unified from the renormalization group (RG) theoretical point of view. However, the recognition has been only formal in the sense that it has not given us any new insight nor provided any new technical advantage over the usual RG approach. With our approach, the proto RG method proposed here, we can clearly show that system reduction is the key to singular perturbation methods. The approach also makes the calculation of singular perturbation results more transparent than the conventional RG approach. Consequently, for example, a consistent and easy RG derivation of the rotational covariant Newell-Whitehead-Segel equation is possible.

Of the three, the first one, is the earliest and is also a good read. Further, it also puts several asymptotic analysis techniques–methods of multiple scales, boundary layers, asymptotic mathcing, WKB–in perspective, and discusses them in the renormalization group (RG) framework. And, the last one gives an algorithm with five steps to obtain RG equations (Hat tip: KuoAn Wu for the pointer).

Some other links of interest: Prof. Goldenfeld’s homepage; Prof. Jonathan A Dantzig’s homepage; Wiki page on renormalization.