Generalized Multistep Predictor-Corrector Methods

ABSTRACT

The order p which is obtainable with a stable k-step method in the numerical solution of y′ = f(x, y) is limited to p = k + 1 by the theorems of Dahlquist. In the present paper the customary schemes are modified by including the value of the derivative at one “nonstep point;” as usual, this value is gained from an explicit predictor. It is shown that the order of these generalized predictor-corrector methods is not subject to the above restrictions; stable k-step schemes with p = 2k + 2 have been constructed for k ≤ 4. Furthermore it is proved that methods of order p actually converge like hp uniformly in a given interval of integration. Numerical examples give some first evidence of the power of the new methods.

REFERENCES

• 1: dahlquistT, G. Convergmlee and stability in the numerical integration of ordinary differential equations. Math. Scand, 4 (1956), 33-53.
• 2: HENRIVCI, P. Discrete variable methods in ordinary differential equations. Wiley, 1962.
• 3: UTRABE:, M., YANAWANA, H., AND SmNOmaRA, Y. Periodic solutions of van der Pol's eqmtion with damping eoeflieient X = 2 10. J. Sci. Hiroshima Univ. {A}, 23 (1960), 325-366. See also UaABE, M. Theory of errors in numerical integration of ordinary differential equations, Teeh. Rep. 183, U. Wisconsin Math. Res. Center, 1960, 88p.
• 4: SERTTER, H .J . On the convergence of characteristic finite-difference methods of high accuracy for quasi-linear hyperbolic equations. Numer. Math. 3 (1961), 321-344.
• 5: T. E. Hull , A. L. Creemer, Efficiency of Predictor-Corrector Procedures, Journal of the ACM (JACM), v.10 n.3, p.291-301, July 1963 [doi>10.1145/321172.321176]
• 6: Begnaud Francis Hildebrand, Introduction to numerical analysis: 2nd edition, Dover Publications, Inc., New York, NY, 1987

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.7 Ordinary Differential Equations
          Subjects: Multistep and multivalue methods

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS

General Terms:
Algorithms, Measurement, Performance, Reliability, Theory, Verification