Second Derivative Block Hybrid Methods for the Numerical Integration of Differential Systems

Abstract

The second derivative block hybrid method for the continuous integration of differential systems within the interval of integration was derived. The second derivative block hybrid method maintained the stability properties of the Runge–Kutta methods suitable for solving stiff differential systems. The lack of such stability properties makes the continuous solution not reliable, especially in solving large stiff differential systems. We derive these methods by using one intermediate off-grid point in between the familiar grid points for continuous solution within the interval of integration. The new family had a high accuracy, non-overlapping piecewise continuous solution with very low error constants and converged under the suitable conditions of stability and consistency. The results of computational experiments are presented to demonstrate the efficiency and usefulness of the methods, which also indicate that the block hybrid methods are competitive with some strong stability stiff integrators.

Comparing the Performance of Two Butcher-Based Block Hybrid Algorithms for the Solution of Initial Value Problems

“…However, this resulted in a higher condition number than both y n+ 7 4 and y n+ 5 2 . On a final note, the results of both methods presented in this study outperform the second-order and fourteenth-order second derivative method of [35] with less function evaluations and computational time.…”

“…Table 3 showed that the block hybrid method y n+ 5 2 performed better than y n+ 7 4 using the same initial conditions, especially for x = 50. Nevertheless, both methods, though of order five, outperformed Yakubu et al's [35] fourteenth-order method, as summarized in Table 4. Figure 3, which is a log-log plot of the errors and step sizes, showed that, while both methods performed at par for y 1 (x), the same was not the case for y 2 (x) because y n+ 7 4 performed better than y n+ 5 2 .…”

“…Table 7 and Figure 6 showed that the block hybrid method y n+ 5 2 performed at par with those of y n+ 7 4 using the same initial conditions. Nevertheless, both methods, though of order five, outperformed Yakubu et al's [35] fourteenth-order method, as shown in Table 8.…”

“…We used a 64-bit DELL Latitude laptop running on Intel(R) Core(TM) i5-5200U CPU @ 2.20GHz, manufactured in the USA, in all numerical computations. The first and last test problems were drawn from [35] with the sole aim of comparing the results of our method with theirs. The second test problem was from Chemistry.…”

“…Throughout this section, we used a constant step size of h = 0.1 on the same integration interval of [0, 50], as shown in Tables 3-7 and their respective figures. However, for the purpose of comparison with the work in [35], we extended the interval to [0, 500], as shown in Tables 4 and 8.…”