We present a novel way of constructing reduced models for systems of ordinary
differential equations. 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.
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 this way
we are able 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. Under suitable assumptions,
one can calculate the higher order terms by a simple and efficient recursive
algorithm.
Keywords
model reduction, Mori–Zwanzig, renormalization,
singularity, partial differential equations
