On the overall stability and convergence of single-step integration schemes for ordinary differential equations

  • Authors:

  • John W. Carr, III

  • Affiliations:

  • -

  • Venue:
  • ACM '56 Proceedings of the 1956 11th ACM national meeting
  • Year:
  • 1956

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Abstract

Although numerous writers have stated that the class of single-step (“Runge-Kutta”) methods of numerical integration of ordinary differential equations is stable under calculation or round-off error, no one has given formal equations for the bounds on the propagated error to indicate this stability. Rutishauser (1) justifies the stability by noting that there is only one solution to the approximating differential equation, and Hildebrand (2) calculates a propagated error bound for the simplest (Euler) case. However the latter bound does not indicate the stability for even that case.