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Consistency, convergence and stability of general discretizations of the initial value problem

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Abstract

Discretizations of nonlinear operators in Banach space are described and the concept of an “inverse” discretization introduced. In the main part of the paper, the very general formalism of BUTCHER for the initial value problem for ordinary differential equations is examined and the sufficiency of conditions for its stability and convergence is demonstrated. The order of convergence of these methods is discussed, and an example is given.

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References

  1. Butcher, J. C.: On the convergence of numerical solutions to ordinary differential equations. Mathematics of Computation20, 1–10 (1966).

    Google Scholar 

  2. England, R.: Automatic methods for solving systems of ordinary differential equations. Thesis, University of Liverpool (1967).

  3. Henrici, P.: Discrete variable methods for ordinary differential equations. New York: John Wiley 1962.

    Google Scholar 

  4. ——: Error propagation for difference methods. New York: John Wiley 1963

    Google Scholar 

  5. Pereyra, V.: Iterated deferred corrections for nonlinear operator equations. Numerische Mathematik10, 316–323 (1967)

    Google Scholar 

  6. Stetter, H. J.: Asymptotic expansions for the error of discretization algorithms for non-linear functional equations. Numerische Mathematik7, 18–31 (1965)

    Google Scholar 

  7. ——: Stability of nonlinear discretization algorithms, inJ. H. Bramble (ed.), Numerical solution of partial differential equations. New York: Academic Press 1966.

    Google Scholar 

  8. ——: A study of strong and weak stability in discretization algorithms. J. SIAM Numer. and Ser. B2, 265–280 (1965).

    Google Scholar 

  9. Watt, J. M.: The asymptotic discretization error of a class of methods for solving ordinary differential equations. Proc. Camb. Phil. Soc.63, 461–472 (1967)

    Google Scholar 

  10. ——: Convergence and stability of discretization methods for functional equations. Computer Journal11, 77–82 (1968).

    Google Scholar 

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Authors and Affiliations

  1. Department of Computational and Statistical Science, University of Liverpool, 3, Liverpool, Great Britain

    J. M. Watt

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  1. J. M. Watt
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Watt, J.M. Consistency, convergence and stability of general discretizations of the initial value problem. Numer. Math. 12, 11–22 (1968). https://doi.org/10.1007/BF02170992

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  • DOI: https://doi.org/10.1007/BF02170992

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