Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
SIAM Journal on Numerical Analysis
A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations
ACM Transactions on Mathematical Software (TOMS)
Algorithm 407: DIFSUB for solution of ordinary differential equations [D2]
Communications of the ACM
Explicit, Time Reversible, Adaptive Step Size Control
SIAM Journal on Scientific Computing
Linearly-implicit two-step methods and their implementation in Nordsieck form
Applied Numerical Mathematics - The third international conference on the numerical solutions of volterra and delay equations, May 2004, Tempe, AZ
Time-step selection algorithms: adaptivity, control, and signal processing
Applied Numerical Mathematics - The third international conference on the numerical solutions of volterra and delay equations, May 2004, Tempe, AZ
The best parameterization for parametric interpolation
Journal of Computational and Applied Mathematics - Special issue: The international conference on computational methods in sciences and engineering 2004
Nordsieck methods on nonuniform grids: stability and order reduction phenomenon
Mathematics and Computers in Simulation
LSODE and LSODI, two new initial value ordinary differnetial equation solvers
ACM SIGNUM Newsletter
Constant coefficient linear multistep methods with step density control
Journal of Computational and Applied Mathematics
Superconvergent explicit two-step peer methods
Journal of Computational and Applied Mathematics
Doubly quasi-consistent parallel explicit peer methods with built-in global error estimation
Journal of Computational and Applied Mathematics
Variable-Stepsize Interpolating Explicit Parallel Peer Methods with Inherent Global Error Control
SIAM Journal on Scientific Computing
Global Error Control in Adaptive Nordsieck Methods
SIAM Journal on Scientific Computing
Variable-stepsize doubly quasi-consistent parallel explicit peer methods with global error control
Applied Numerical Mathematics
Local and global error estimation and control within explicit two-step peer triples
Journal of Computational and Applied Mathematics
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In this paper we study advantages of numerical integration by quasi-consistent Nordsieck formulas. All quasi-consistent numerical methods possess at least one important property for practical use, which has not attracted attention yet, i.e. the global error of a quasi-consistent method has the same order as its local error. This means that the usual local error control will produce a numerical solution for the prescribed accuracy requirement if the principal term of the local error dominates strongly over remaining terms. In other words, the global error control can be as cheap as the local error control in the methods under discussion. Here, we apply the above-mentioned idea to Nordsieck-Adams-Moulton methods, which are known to be quasi-consistent. Moreover, some Nordsieck-Adams-Moulton methods are even super-quasi-consistent. The latter property means that their propagation matrices annihilate two leading terms in the defect expansion of such methods. In turn, this can impose a strong relation between the local and global errors of the numerical solution and allow the global error to be controlled effectively by a local error control. We also introduce Implicitly Extended Nordsieck methods such that in some sense they form pairs of embedded formulas with their source Nordsieck methods. This facilitates the local error control in quasi-consistent Nordsieck schemes. Numerical examples presented in this paper confirm clearly the power of quasi-consistent integration in practice.