William B. Gragg
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.
- 1 dahlquistT, G. Convergmlee and stability in the numerical integration of ordinary differential equations. Math. Scand, 4 (1956), 33-53.Google Scholar
- 2 HENRIVCI, P. Discrete variable methods in ordinary differential equations. Wiley, 1962.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 5 HULL, T. E., AND CREMER, A. L. Efficiency of predictoeorreetor procedures. J ACM 10 (1963), 291-301. Google Scholar
- 6 HILDEBRAND, F .B . fntroduction to Numerical Analysis. McGraw-Hill, 1956. Google Scholar
Index Terms
- Generalized Multistep Predictor-Corrector Methods
Recommendations
Predictor---Corrector Methods for General Regularized Nonconvex Variational Inequalities
This paper is devoted to the study of a new class of nonconvex variational inequalities, named general regularized nonconvex variational inequalities. By using the auxiliary principle technique, a new modified predictor---corrector iterative algorithm ...
Predictor-corrector iterative algorithms for solving generalized mixed quasi-variational-like inclusion
By applying the concept of partially relaxed @h-strong monotonicity of set-valued mappings due to author and the auxiliary variational inequality technique, some new predictor-corrector iterative algorithms for solving generalized mixed quasi-...
Predictor-corrector iterative algorithms for solving generalized mixed variational-like inequalities
The concept of partially relaxed @h-strong monotonicity of set-valued mappings is introduced. By applying the auxiliary variational inequality technique, some new predictor-corrector iterative algorithms for solving generalized mixed variational-like ...
Comments