Two FORTRAN packages for assessing initial value methods
ACM Transactions on Mathematical Software (TOMS)
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Iterative schemes for three-stage implicit Runge-Kutta methods
Applied Numerical Mathematics
Starting algorithms for IRK methods
Journal of Computational and Applied Mathematics
On the numerical solution of stiff IVPs by Lobatto IIIA Runge-Kutta methods
ICCAM '96 Proceedings of the seventh international congress on Computational and applied mathematics
Are high order variable step equistage initializers better than standard starting algorithms?
Journal of Computational and Applied Mathematics
Speeding up Netwton-type iterations for stiff problems
Journal of Computational and Applied Mathematics
Stage value predictors for additive and partitioned Runge-Kutta methods
Applied Numerical Mathematics
A time-adaptive finite volume method for the Cahn-Hilliard and Kuramoto-Sivashinsky equations
Journal of Computational Physics
Stage value predictors for additive and partitioned Runge--Kutta methods
Applied Numerical Mathematics
Speeding up Newton-type iterations for stiff problems
Journal of Computational and Applied Mathematics
Efficient iterations for Gauss methods on second-order problems
Journal of Computational and Applied Mathematics
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This paper deals with starting algorithms for Newton-type schemes for solving the stage equations of implicit s-stages Runge-Kutta methods applied to stiff problems. We present a family of starting algorithms with orders from 0 to s + 1 and, with estimations of the error in these algorithms, we give a technique for selecting, at each step, the most convenient in the family. The proposed algorithms, that can be expressed in terms of divided differences, are based on the Lagrange interpolation of the stages of the last two integration steps. We also analyse the orders of the starting algorithms for the non-stiff case, for the Prothero and Robinson model and the stiff order. Finally, by means of some numerical experiments we show that this technique allows, in general, to greatly improve the performance of implicit Runge-Kutta methods on stiff problems.