We present two pairs of embedded Runge-Kutta type methods for direct solution of fourth-order ordinary differential equations (ODEs) of the form
This paper deals with embedded RKFD methods for directly solving special fourth-order ordinary differential equations (ODEs) of the form
The general form of RKFD method with
The parameters
To determine the parameters of the RKFD method given in (
In this paper we will derive embedded Runge-Kutta pairs for direct integration of special fourth-order ODEs. Embedded pairs of RK type methods have a built-in local truncation error estimate; as a result, the step size can be controlled at virtually no extra cost, and hence an efficient variable step size code can be developed.
In recent years, the construction of embedded Runge-Kutta method is an effective research area yielding continuous development to the existing codes. The present paper is primarily dedicated as an extra work in this research area. This technique involves two Runge-Kutta formulae of orders
The remainder of this paper is organized as follows. In Section
The order conditions of RKFD method up to fifth order have been derived using Taylor series expansion by Hussain et al. [
The order conditions for Fourth order: Fifth order: Sixth order: Seventh order:
The order conditions for Third order: Fourth order: Fifth order: Sixth order: Seventh order:
The order conditions for Second order: Third order: Fourth order: Fifth order: Sixth order: Seventh order:
The order conditions for First order: Second order: Third order: Fourth order: Fifth order: Sixth order: Seventh order: The quantities of where The following quantities given in [ where We defined the local error estimation at the point where where
The following strategies are utilized for developing efficient embedded pairs.
The local error estimation, EST, can be used to control the step size
If
The construction of embedded explicit RKFD pairs will be discussed in this section. In specific, we will derive two embedded RKFD pairs of orders 5(4) and 6(5) with three and four stages per step, respectively.
This section will focus on the derivation of embedded RKFD5(4) pair with three stages. The authors in [
The graph of
The quantities
In this section a four-stage embedded RKFD6(5) pair will be derived. For the sixth-order
The graph of
Finally, all the parameters of the embedded pair are represented in the Butcher tableau and denoted as RKFD6
In this section, we present some problems which involve special fourth-order ODEs of the form RKFD6(5): the new embedded Runge-Kutta pair of orders 6(5) derived in Section RKFD5(4): the new embedded Runge-Kutta pair of orders 5(4) derived in Section RK6(5)V: the embedded Runge-Kutta pair of orders 6(5) derived by Verner as given in [ RK5(4)D: the embedded Runge-Kutta pair of orders 5(4) with FSAL property derived by Dormand and Prince [
We consider that the homogeneous linear problem is as follows:
We consider that the homogeneous nonlinear problem is as follows:
We consider that the inhomogeneous nonlinear problem is as follows:
We consider that the linear system is as follows:
In the numerical experiments, we have computed for each method and problem the maximum global error and the number of function evaluations used in the integration. In Figures
The comparison results of the total number of successful steps and failure steps between RKFD6(
Problems | Methods | Successful steps | Failure steps |
---|---|---|---|
1 | RKFD6( |
440 | 4 |
RKFD5( |
705 | 0 | |
RK6( |
461 | 9 | |
RK5( |
768 | 0 | |
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2 | RKFD6( |
151 | 0 |
RKFD5( |
200 | 0 | |
RK6( |
208 | 0 | |
RK5( |
253 | 0 | |
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3 | RKFD6( |
330 | 4 |
RKFD5( |
466 | 0 | |
RK6( |
348 | 8 | |
RK5( |
497 | 0 | |
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4 | RKFD6( |
264 | 5 |
RKFD5( |
340 | 6 | |
RK6( |
339 | 5 | |
RK5( |
484 | 5 |
The efficiency curves for Problem
The efficiency curves for Problem
The efficiency curves for Problem
The efficiency curves for Problem
We have constructed the two pairs of embedded RKFD methods for directly solving special fourth-order ODEs of the form
The authors declare that there is no conflict of interests regarding the publication of this paper.