Stable time filtering of strongly unstable spatially extended systems

Marcus J. Grote, Andrew J. Majda

Research output: Contribution to journalArticlepeer-review


Many contemporary problems in science involve making predictions based on partial observation of extremely complicated spatially extended systems with many degrees of freedom and with physical instabilities on both large and small scale. Various new ensemble filtering strategies have been developed recently for these applications, and new mathematical issues arise. Because ensembles are extremely expensive to generate, one such issue is whether it is possible under appropriate circumstances to take long time steps in an explicit difference scheme and violate the classical Courant-Friedrichs-Lewy (CFL)-stability condition yet obtain stable accurate filtering by using the observations. These issues are explored here both through elementary mathematical theory, which provides simple guidelines, and the detailed study of a prototype model. The prototype model involves an unstable finite difference scheme for a convection-diffusion equation, and it is demonstrated below that appropriate observations can result in stable accurate filtering of this strongly unstable spatially extended system.

Original languageEnglish (US)
Pages (from-to)7548-7553
Number of pages6
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number20
StatePublished - May 16 2006


  • Courant-Friedrichs-Lewy condition
  • Kalman filter
  • Observability
  • Stable filter
  • Unstable finite differences

ASJC Scopus subject areas

  • General


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