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Functionally fitted explicit pseudo two-step Runge-Kutta methods. (English) Zbl 1158.65050
The paper presents a new family of functionally fitted explicit pseudo two-step Runge-Kutta (EPTRK) methods aimed at integrating an equation exactly if its solution is a linear combination of a chosen set of basic functions.
The first section is an introduction concerning Runge-Kutta methods (implicit, explicit and explicit pseudo two-step) for numerical solving of a Cauchy problem.
The second section briefly reviews EPTRK methods and outlines their main strengths and weakness.
In the third section the authors detail how to derive the new functionally fitted variants of Runge-Kutta methods, called FEPTRK and establish under which condition it is possible to do so.
The fourth section concerns the accuracy and stability properties of these variants. The main results are than an \(s\)-stage FEPTRK which has a local accuracy order of \(s\) in general and \(s+2\) in some cases.
Section five discusses enhancements important to producing competitive numerical codes, namely variable stepsize considerations, error estimation via embedded methods, as well as continuous extensions.
Numerical experiments are presented in the sixth section. The authors compare the new functionally fitted methods with with some classical ones, indicating the efficiency of the new methods.
The last section gives some concluding remarks.

MSC:
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems, general theory
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
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