Acta mathematica scientia,Series B ›› 2010, Vol. 30 ›› Issue (2): 428-446.doi: 10.1016/S0252-9602(10)60057-4

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AN ADAPTIVE VERSION OF GLIMM'S SCHEME

 H. Kim, M. Laforest, D. Yoon   

  1. 1.Department of Mathematics, Korea University, Anam-dong, Sungbuk-ku, Seoul, 136-701, Korea;
    2.Département de mathématiques et génie industriel, École Polytechnique de Montréal, Montréal, Québec, H3C 3A7, Canada
  • Received:2009-11-17 Online:2010-03-20 Published:2010-03-20
  • Supported by:

    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.

Abstract:

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 ad-hoc. 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.

Key words: conservation laws, finite difference methods, adaptive, error estimation, a-posteriori

CLC Number: 

  • 65M06
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