Problems with multiple temporal and spatial scales

M. Brio, G. M. Webb, A. R. Zakharian

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

Problems with multiple disparate temporal and spatial scales are at the heart of most physical problems, as well as in mathematical modeling problems, in economics and social sciences. Numerical treatment usually allows one to deal with two, or at most, three disparate scales due to limitations of computer resources. Problems with different scales arise when the interest is in the description of the problem on the larger scales, for example the motion of weather fronts in the presence of fast gravity waves; the average trajectory of a fast rotating charged particle in a magnetic field, in which one averages over the fast gyro-period of the particle; the behavior of society incorporating the dynamics of individuals; the role of singularities in solutions of differential equations or the role of geometry; etc. The methods of dealing with such problems are based on either resolving (if ever possible) or not resolving the fine scales. In the latter case, asymptotic analysis of the dominant terms is often used to separate scales or compute the cumulative effect of the small scale motion on the large scale dynamics. For problems where scales are weakly coupled, numerical treatment may often produce physical results as long as the choice of the method has remnants of the smaller scale behavior such as: damping, diffusion, dispersion, etc. For the second type of problem, under-resolved numerical computations usually never produce physical results as cumulative effects are not of the form of the truncation errors, in addition to the fact that the details of the small scale motion are important and have to be accounted for on either a theoretical or experimental basis. In this Chapter we consider examples from stiff ODEs, long-time integrators, Hamiltonian systems, multi-symplectic systems, hyperbolic conservation laws, Godunov methods, Riemann solvers with slope/flux limiters.

Original languageEnglish (US)
Title of host publicationMathematics in Science and Engineering
PublisherElsevier
Pages175-249
Number of pages75
EditionC
DOIs
StatePublished - Jan 1 2010
Externally publishedYes

Publication series

NameMathematics in Science and Engineering
NumberC
Volume213
ISSN (Print)0076-5392

Keywords

  • Flux limiters
  • Godunov methods
  • Hamiltonian systems
  • Hyperbolic conservation laws
  • Long-time integrators
  • Multi-symplectic systems
  • Multiple scales methods
  • Riemann solvers
  • Stiff ODEs

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)

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  • Cite this

    Brio, M., Webb, G. M., & Zakharian, A. R. (2010). Problems with multiple temporal and spatial scales. In Mathematics in Science and Engineering (C ed., pp. 175-249). (Mathematics in Science and Engineering; Vol. 213, No. C). Elsevier. https://doi.org/10.1016/S0076-5392(10)21310-3