Dynamics of Partial Differential Equations

Volume 8 (2011)

Number 1

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

Pages: 21 – 45

DOI: https://dx.doi.org/10.4310/DPDE.2011.v8.n1.a3

Authors

Jerrold E. Marsden (Control and Dynamical Systems, California Institute of Technology, Pasadena, Calif.)

Houman Owhadi (Control and Dynamical Systems, California Institute of Technology, Pasadena, Calif.)

Molei Tao (Control and Dynamical Systems, California Institute of Technology, Pasadena, Calif.)

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 [32] 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.

Keywords

stiff PDEs, FLow AVeraging integratORS

2010 Mathematics Subject Classification

35-xx, 65-xx

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Published 7 April 2011