Choose inter-element coupling to preserve self-adjoint dynamics in multiscale modelling and computation

Authors:
A. J. Roberts
Affiliations:
School of Mathematical Sciences, University of Adelaide, South Australia 5005, Australia
Venue:
Applied Numerical Mathematics
Year:
2010

Citing 13
Cited 0

Nonlinear Galerkin methods

SIAM Journal on Numerical Analysis
The utility of an invariant manifold description of the evolution of a dynamical system

SIAM Journal on Mathematical Analysis
A centre manifold description of containment dispersion in channels with varying flow properties

SIAM Journal on Applied Mathematics
Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations

Physica D
Weakly nonlinear stability analyses of prototype reaction-diffusion model equations

SIAM Review
Elements of applied bifurcation theory (2nd ed.)

Elements of applied bifurcation theory (2nd ed.)
Space-Time Continuous Analysis of Waveform Relaxation for the Heat Equation

SIAM Journal on Scientific Computing
New high-resolution central schemes for nonlinear conservation laws and convection—diffusion equations

Journal of Computational Physics
A multigrid tutorial: second edition

A multigrid tutorial: second edition
Holistic discretization ensures fidelity to Burgers' equation

Applied Numerical Mathematics
A holistic finite difference approach models linear dynamics consistently

Mathematics of Computation
Subgrid Upscaling and Mixed Multiscale Finite Elements

SIAM Journal on Numerical Analysis
General Tooth Boundary Conditions for Equation Free Modeling

SIAM Journal on Scientific Computing

Quantified Score

Hi-index	0.00

Visualization

Abstract

Consider the macroscale modelling of microscale spatio-temporal dynamics. Here we develop an approach to ensure coarse scale discrete models preserve important self-adjoint properties of the microscale dynamics. The first part explores the discrete modelling of microscale continuum dynamics in multiple spatial dimensions. The second part addresses how dynamics on a fine lattice are mapped to lattice a factor of two coarser (as in multigrids); for simplicity we address only one-dimensional lattices. Such mapping of discrete lattice dynamics may be iterated to empower future research to explore scale dependent emergent phenomena. The support of the dynamical systems theory of centre manifolds ensures that the coarse scale modelling applies with a finite spectral gap, in a finite domain, and for all time. The accuracy of the modelling is limited by the asymptotic resolution of subgrid scale processes. As given examples demonstrate, the approach developed here ensures the preservation of important symmetries of the microscale dynamics.