A new family of multistep numerical integration methods based on Hermite interpolation

Abstract

In this paper, a new family of explicit and implicit multistep methods is presented both for the error-controlled and uncontrolled modes. The main concept is to replace the Newton interpolation with the Hermite interpolation, where the Hermite polynomial is fitted to the function values and its derivatives. This idea is very useful in the numerical solution of problems (e.g., orbit propagation problem) where higher-order derivatives can easily be computed. In addition to the theoretical concept, the stability regions of the proposed methods are determined. The new methods are more stable than the well-known multistep numerical integrators (i.e., Adams–Bashforth and Adams–Bashforth–Moulton) in the explicit, implicit, and predictor–corrector forms. Using the second-order derivatives gives smaller error constants in the proposed method. The new integrators are numerically tested for a few examples, and the solutions are compared with those of the well-known multistep methods. Moreover, the CPU time and absolute integration error are compared in the satellite orbit propagation problem using various integration methods. The CHAMP mission, i.e., a German small-satellite mission for geoscientific and atmospheric research and applications, is considered as a case study for comparing the achievable accuracy of the proposed method with the existing method for solving the two-body problem.

Appendix

1.1 Backward difference

The difference approximation could be achieved using Taylor series. If $f_{k-1}$ and ${f}^{\prime }_{k-1}$ are expanded to Taylor series, then we have

$$\begin{aligned} \begin{aligned} \hbox {I)}\quad f_{k-1}&=f(t_k -h)=f_k -h.{f}^{\prime }_k +\frac{h^{2}}{2!}f_k^{(2)} -\frac{h^{3}}{3!}f_k^{(3)} +\frac{h^{4}}{4!}f_k^{(4)} +\cdots \\ \hbox {II)}\quad {f}^{\prime }_{k-1}&={f}'(t_k -h)={f}^{\prime }_k -h.f_k^{(2)} +\frac{h^{2}}{2!}f_k^{(3)} -\frac{h^{3}}{3!}f_k^{(4)} +\cdots , \end{aligned} \end{aligned}$$

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where

$$\begin{aligned} \begin{aligned} f_k&=f(t_k ) \\ {f}^{\prime }_k&=\frac{\partial f(t)}{\partial t}\left| {_{t=t_k}}\right. \end{aligned} \end{aligned}$$

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The $f_k^{(2)}$ could be computed from the following equation:

$$\begin{aligned} 3f_{k-1} +h.{f}^{\prime }_{k-1}&= 3f_k -2h.{f}^{\prime }_k +\frac{h^{2}}{2}f_k^{(2)} -\frac{h^{4}}{24}f_k^{(4)} \nonumber \\ \Rightarrow f_k^{(2)}&= \frac{2}{h^{2}}\left[ {3f_{k-1} +h.{f}^{\prime }_{k-1} -3f_k +2h.{f}^{\prime }_k +\frac{h^{4}}{24}f_k^{(4)}}\right] . \end{aligned}$$

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Similarly, $f_k^{(3)}$

$$\begin{aligned} f_{k-1} +\frac{h}{2}.{f}^{\prime }_{k-1}&= f_k -\frac{h}{2}.{f}^{\prime }_k +\frac{h^{3}}{12}f_k^{(3)} -\frac{h^{4}}{12}f_k^{(4)} \nonumber \\ \Rightarrow f_k^{(3)}&= \frac{6}{h^{3}}\left[ {2f_{k-1} +h.{f}^{\prime }_{k-1} -2f_k +h.{f}^{\prime }_k +\frac{h^{4}}{6}f_k^{(4)} } \right] . \end{aligned}$$

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