Diagonally implicit Runge-Kutta-Nystro¨m methods for oscillatory problems
SIAM Journal on Numerical Analysis
Robust defect control with Runge-Kutta schemes
SIAM Journal on Numerical Analysis
The tolerance proportionality of adaptive ODE solvers
Journal of Computational and Applied Mathematics - Special issue on numerical methods for ordinary differential equations
Reliability of local error control algorithms for initial value ordinary differential equations
Proceedings of the IFIP TC2/WG2.5 working conference on Quality of numerical software: assessment and enhancement
Numerical investigations on global error estimation for ordinary differential equations
ICCAM '96 Proceedings of the seventh international congress on Computational and applied mathematics
A Code Based on the Two-Stage Runge-Kutta Gauss Formula for Second-Order Initial Value Problems
ACM Transactions on Mathematical Software (TOMS)
A Code Based on the Two-Stage Runge-Kutta Gauss Formula for Second-Order Initial Value Problems
ACM Transactions on Mathematical Software (TOMS)
Local and global error estimation and control within explicit two-step peer triples
Journal of Computational and Applied Mathematics
Hi-index | 7.29 |
Modern codes for the numerical solution of Initial Value Problems (IVPs) in ODEs are based in adaptive methods that, for a user supplied tolerance @d, attempt to advance the integration selecting the size of each step so that some measure of the local error is ~@d. Although this policy does not ensure that the global errors are under the prescribed tolerance, after the early studies of Stetter [Considerations concerning a theory for ODE-solvers, in: R. Burlisch, R.D. Grigorieff, J. Schroder (Eds.), Numerical Treatment of Differential Equations, Proceedings of Oberwolfach, 1976, Lecture Notes in Mathematics, vol. 631, Springer, Berlin, 1978, pp. 188-200; Tolerance proportionality in ODE codes, in: R. Marz (Ed.), Proceedings of the Second Conference on Numerical Treatment of Ordinary Differential Equations, Humbold University, Berlin, 1980, pp. 109-123] and the extensions of Higham [Global error versus tolerance for explicit Runge-Kutta methods, IMA J. Numer. Anal. 11 (1991) 457-480; The tolerance proportionality of adaptive ODE solvers, J. Comput. Appl. Math. 45 (1993) 227-236; The reliability of standard local error control algorithms for initial value ordinary differential equations, in: Proceedings: The Quality of Numerical Software: Assessment and Enhancement, IFIP Series, Springer, Berlin, 1997], it has been proved that in many existing explicit Runge-Kutta codes the global errors behave asymptotically as some rational power of @d. This step-size policy, for a given IVP, determines at each grid point t"n a new step-size h"n"+"1=h(t"n;@d) so that h(t;@d) is a continuous function of t. In this paper a study of the tolerance proportionality property under a discontinuous step-size policy that does not allow to change the size of the step if the step-size ratio between two consecutive steps is close to unity is carried out. This theory is applied to obtain global error estimations in a few problems that have been solved with the code Gauss2 [S. Gonzalez-Pinto, R. Rojas-Bello, Gauss2, a Fortran 90 code for second order initial value problems, ], based on an adaptive two stage Runge-Kutta-Gauss method with this discontinuous step-size policy.