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1 results

  • 20 - Linear Multi-step Methods

  • from Part IV - Initial Value Problems for Ordinary Differential Equations
  • Abner J. Salgado, University of Tennessee, Knoxville, Steven M. Wise, University of Tennessee, Knoxville
  • Book:
    Classical Numerical Analysis
    Published online:
    29 September 2022
    Print publication:
    20 October 2022, pp 555-580

  • Summary

    We motivate the introduction of multistep schemes by recalling interpolatory quadrature rules. Then, these are presented in the general setting. Their consistency and conditions of consistency: the method of C’s and the log method are discussed. The construction and consistency of Adams-Bashforth, Adams-Moulton, and backward differentiation formulas is then presented. Then, we turn our attention to the delicate issue of stability for multistep schemes: the notions of zero stability, root condition, and homogeneous zero stability are introduced and their equivalence discussed. This is achieved by developing the theory of solutions to linear difference equations. The celebrated Dahlquist equivalence theorem, stating that a linear multistep scheme is convergent if and only if it is consistent and stable is then presented. A discussion of Dahlquist first barrier concludes the chapter