Renormalized reduced models for singular PDEs

Abstract

We present a novel way of constructing reduced models for systems of ordinary differential equations. The reduced models we construct depend on coefficients which measure the importance of the different terms appearing in the model and need to be estimated. The proposed approach allows the estimation of these coefficients on the fly by enforcing the equality of integral quantities of the solution as computed from the original system and the reduced model. In particular, the approach combines the concepts of renormalization and effective field theory developed in the context of high energy physics and the Mori-Zwanzig formalism of irreversible statistical mechanics. It allows to construct stable reduced models of higher order than was previously possible. The method is applied to the problem of computing reduced models for ordinary differential equation systems resulting from Fourier expansions of singular (or near-singular) time-dependent partial differential equations. Results for the 1D Burgers and the 3D incompressible Euler equations are used to illustrate the construction. We also present, for the 1D Burgers and the 3D Euler equations, a simple and efficient recursive algorithm for calculating the higher order terms.