Phase transition approach to detecting singularities of PDEs

TL;DR

This paper introduces a mesh refinement algorithm that detects singularities in time-dependent PDEs by treating them as phase transitions, enabling the calculation of blow-up rates through three different approaches.

Contribution

The paper presents a unified algorithm that applies phase transition concepts to detect singularities and compute blow-up rates in PDEs, integrating three distinct methods.

Findings

Successfully applied to Burgers and Schrödinger equations

Demonstrated accurate blow-up rate calculations

Unified framework for multiple approaches

Abstract

We present a mesh refinement algorithm for detecting singularities of time-dependent partial differential equations. The main idea behind the algorithm is to treat the occurrence of singularities of time-dependent partial differential equations as phase transitions. We show how the mesh refinement algorithm can be used to calculate the blow-up rate as we approach the singularity. This calculation can be done in three different ways: i) the direct approach where one monitors the blowing-up quantity as it approaches the singularity and uses the data to calculate the blow-up rate ; ii) the "phase transition" approach (\`a la Wilson) where one treats the singularity as a fixed point of the renormalization flow equation and proceeds to compute the blow-up rate via an analysis in the vicinity of the fixed point and iii) the "scaling" approach (\`a la Widom-Kadanoff) where one postulates the existence of scaling laws for different quantities close to the singularity, computes the associated exponents and then uses them to estimate the blow-up rate. Our algorithm allows a unified presentation of these three approaches. The inviscid Burgers equation and the supercritical Schrodinger equation are used as instructive examples to illustrate the constructions.