Abstract
The construction of exponentially fitted two-derivative Runge-Kutta (EFTDRK) methods for the numerical solution of first-order differential equations is investigated. The revised EFTDRK methods proposed, with equation-dependent coefficients, take into consideration the errors produced in the internal stages to the update. The local truncation errors and stability of the new methods are analyzed. The numerical results are reported to show the accuracy of the new methods.
Highlights
IntroductionFor the numerical solution of problem (1), classical general-purpose methods such as Runge-Kutta (RK) methods and linear multistep methods (LMMs) can not produce satisfactory results due to the special structure of the problems
We are interested in the numerical integration of initial-value problems (IVPs) of first-order differential equations in the form y (x) = f (x, y), (1)
In contrast to the standard exponential fitting of RK methods, we take into account the contribution of the local error of internal stages to the update
Summary
For the numerical solution of problem (1), classical general-purpose methods such as Runge-Kutta (RK) methods and linear multistep methods (LMMs) can not produce satisfactory results due to the special structure of the problems. The possible ways to construct the numerical methods adapted to the character of the solutions can be obtained by using exponential fitting (EF) technique [3,4,5,6,7,8,9]. In standard approach to deriving exponentially fitted Runge-Kutta(-Nystrom) methods, the effect of the error in the internal stages to the error of the final stage is completely neglected. Runge-Kutta methods with three stages by evaluating the errors in each internal stage.
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