An adaptive version of Glimm's scheme

H. Kim, M. Laforest, D. Yoon

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


This article describes a local error estimator for Glimm's scheme for hyperbolic systems of conservation laws and uses it to replace the usual random choice in Glimm's scheme by an optimal choice. As a by-product of the local error estimator, the procedure provides a global error estimator that is shown numerically to be a very accurate estimate of the error in L1(ℝ) for all times. Although there is partial mathematical evidence for the error estimator proposed, at this stage the error estimator must be considered adhoc. Nonetheless, the error estimator is simple to compute, relatively inexpensive, without adjustable parameters and at least as accurate as other existing error estimators. Numerical experiments in 1-D for Burgers' equation and for Euler's system are performed to measure the asymptotic accuracy of the resulting scheme and of the error estimator.

Original languageEnglish
Pages (from-to)428-446
Number of pages19
JournalActa Mathematica Scientia
Issue number2
Publication statusPublished - 2010 Mar

Bibliographical note

Funding Information:
∗Received November 17, 2009. The work of H. Kim was supported by a Korea Research Foundation Grant from the Korean Government (MOEHRD) (KRF-2007-331-C00053). The work of M. Laforest was supported by the National Science and Engineering Council of Canada and the Canadian Foundation for Innovation.


  • 35L65
  • 65M06
  • 65M15
  • a-posteriori
  • adaptive
  • conservation laws
  • error estimation
  • finite difference methods

ASJC Scopus subject areas

  • General Mathematics
  • General Physics and Astronomy


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