Space-time FLAVORS: finite difference, multisymlectic, and pseudospectral integrators for multiscale PDEs

TL;DR

This paper introduces a new class of multiscale integrators for stiff PDEs that leverage flow averaging, are versatile, nonintrusive, and structure-preserving, enabling efficient and accurate simulations without explicit slow variable identification.

Contribution

The paper develops a general framework for multiscale PDE integrators based on flow averaging, extending FLAVORS from ODEs and SDEs to PDEs with structure-preserving properties.

Findings

Efficient multiscale PDE integration with reduced computational cost.

Versatile method applicable without explicit slow variable identification.

Structure-preserving integrators for Hamiltonian PDEs.

Abstract

We present a new class of integrators for stiff PDEs. These integrators are generalizations of FLow AVeraging integratORS (FLAVORS) for stiff ODEs and SDEs introduced in [Tao, Owhadi and Marsden 2010] with the following properties: (i) Multiscale: they are based on flow averaging and have a computational cost determined by mesoscopic steps in space and time instead of microscopic steps in space and time; (ii) Versatile: the method is based on averaging the flows of the given PDEs (which may have hidden slow and fast processes). This bypasses the need for identifying explicitly (or numerically) the slow variables or reduced effective PDEs; (iii) Nonintrusive: A pre-existing numerical scheme resolving the microscopic time scale can be used as a black box and easily turned into one of the integrators in this paper by turning the large coefficients on over a microscopic timescale and off during a mesoscopic timescale; (iv) Convergent over two scales: strongly over slow processes and in the sense of measures over fast ones; (v) Structure-preserving: for stiff Hamiltonian PDEs (possibly on manifolds), they can be made to be multi-symplectic, symmetry-preserving (symmetries are group actions that leave the system invariant) in all variables and variational.