On explicit two-derivative two-step Runge–Kutta methods

Mukaddes Ökten Turaci; Turgut Öziş

doi:10.1007/s40314-018-0719-y

Computational and Applied Mathematics

November 2018, Volume 37, Issue 5, pp 6920–6954 | Cite as

On explicit two-derivative two-step Runge–Kutta methods

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Mukaddes Ökten Turaci
Turgut Öziş

Article

First Online: 04 October 2018

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Abstract

We introduce a class of methods for the numerical solution of ordinary differential equations. These methods called as two-derivative two-step Runge–Kutta methods are extension of the two-step Runge–Kutta methods in which the second derivative of the solution is included. These methods are a special class of second-derivative general linear methods studied by many authors Butcher et al. (Numer Algorithms 40:415–429, 2005), Abdi et al. (Numer Algorithms 57:149–167, 2011), Okuonghae and Ikhile (Numer Algorithms 67(3):637–654, 2014). The order conditions are derived based on the algebraic theory of Butcher (Mathe Comput 26:79–106, 1972) and the $\mathcal {B}-$series theory Hairer and Wanner (Computing 13:1–15, 1974), in a similar way to Chan and Chan (2006). In this study, special explicit two-derivative two-step Runge–Kutta methods that possess one evaluation of the first derivative and many evaluations of the second derivative per step are introduced. Methods with stages up to five and of order up to eight are presented. The numerical calculations have been performed on some non-stiff and mildly stiff problems and comparisons have been made with the accessible methods in the literature.

Keywords

Two-step Runge–Kutta methods Second-derivative general linear methods Two-derivative Runge–Kutta methods Rooted trees Order conditions

Mathematics Subject Classification

65L05 65L06 65L20

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Notes

Acknowledgements

The authors would like to thank the referees and the editor for their valuable comments and suggestions which improved the presentation of the paper.

Appendix A

In this appendix, the values of $\eta (t)$ and $\eta _1(t)$ for trees up to order 5 and order conditions up to order 6 are given. We have assumed $Ae=c$ and $\hat{A}e=\hat{c}$ in the computation values of $\eta (t)$ and $\eta _1(t)$, which result in the stage values to be approximations of order at least one.

$$\begin{aligned} \eta (t)= & {} c,\; \eta _1(t)=c-e,\\ \alpha (t)= & {} \theta E^{-1}(t)+v^T\eta (\emptyset )+w^T\eta _1(\emptyset ).\\ \end{aligned}$$

Order condition:

$-\theta +v^Te+w^Te=1.$

Order condition:

$$\begin{aligned} \frac{1}{2}\theta +v^Tc+w^Tc-w^Te+\hat{v}^Te+\hat{w}^Te=\frac{1}{2}. \end{aligned}$$

Order condition:

$-\frac{1}{3}\theta +v^Tc^2+w^Tc^2-2w^Tc+w^Te+2\hat{v}^Tc+2\hat{w}^Tc-2 \hat{w}^Te=\frac{1}{3}$.

Order condition:

$-\frac{1}{6}\theta +v^TAc+v^T\hat{c}+\frac{1}{2}w^Te+w^TAc-w^Tc+w^T \hat{c}+\hat{v}^Tc+\hat{w}^Tc-\hat{w}^Te=\frac{1}{6}$.

Order condition:

$\frac{1}{4}\theta +v^Tc^3+w^Tc^3-3w^Tc^2+3w^Tc-w^Te+3\hat{v}^Tc^2+3 \hat{w}^Tc^2-6\hat{w}^Tc+3\hat{w}^Te=\frac{1}{4}$.

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Order condition:

$$\begin{aligned} \frac{1}{8}\theta&+v^TcAc+v^Tc\hat{c}+\frac{3}{2}w^Tc-\frac{1}{2} w^Te+w^T(cAc)-w^Tc^2-w^TAc+w^Tc\hat{c}\\&-w^T\hat{c}+\hat{v}^TAc+\hat{v}^T\hat{c}+\hat{v}^Tc^2+\frac{3}{2} \hat{w}^Te+\hat{w}^TAc-3\hat{w}^Tc+\hat{w}^T\hat{c}+\hat{w}^Tc^2=\frac{1}{8}. \end{aligned}$$

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Order condition:

$$\begin{aligned} \frac{1}{12}\theta&+v^TAc^2+2v^T\hat{A}c-\frac{1}{3}w^Te+w^TAc^2 -2w^TAc+w^Tc+2w^T\hat{A}c-2w^T\hat{c}\\&+\hat{v}^Tc^2+\hat{w}^Tc^2-2\hat{w}^Tc+\hat{w}^Te=\frac{1}{12}. \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{24}\theta&+v^TA^2c+v^TA\hat{c}+v^T\hat{A}c-\frac{1}{6}w^Te +\frac{1}{2}w^Tc+w^TA^2c-w^TAc+w^TA\hat{c}+w^T\hat{A}c\\&-w^T\hat{c}+\hat{v}^TAc+\hat{v}^T\hat{c}+\frac{1}{2}\hat{w}^Te +\hat{w}^TAc-\hat{w}^Tc+\hat{w}^T\hat{c}=\frac{1}{24}. \end{aligned}$$

Order condition:

$$\begin{aligned} -\frac{1}{5}\theta&+v^Tc^4+w^Tc^4-4w^Tc^3+6w^Tc^2-4w^Tc+w^Te+4 \hat{v}^Tc^3+4\hat{w}^Tc^3-12\hat{w}^Tc^2\\&+12\hat{w}^Tc-4\hat{w}^Te=\frac{1}{5}. \end{aligned}$$

Order condition:

$$\begin{aligned} -\frac{1}{10}\theta&+v^Tc^2Ac+v^Tc^2\hat{c}+\frac{5}{2}w^Tc^2-2w^Tc +\frac{1}{2}w^Te+w^Tc^2Ac-2w^TcAc+w^TAc\\&-w^Tc^3+w^Tc^2\hat{c}-2w^Tc\hat{c}+w^T\hat{c}+2\hat{v}^TcAc+2\hat{v}^Tc \hat{c}+\hat{v}^Tc^3+6\hat{w}^Tc\\&-2\hat{w}^Te+2\hat{w}^TcAc-2\hat{w}^TAc-5\hat{w}^Tc^2+2\hat{w}^Tc \hat{c}-2\hat{w}^T\hat{c}+\hat{w}^Tc^3=\frac{1}{10}. \end{aligned}$$

Order condition:

$$\begin{aligned} -\frac{1}{15}\theta&+v^TcAc^2+2v^Tc\hat{A}c-\frac{4}{3}w^Tc +\frac{1}{3}w^Te+w^TcAc^2-2w^TcAc+w^Tc^2-w^TAc^2\\&+2w^TAc+2w^Tc\hat{A}c-2w^Tc\hat{c}-2w^T\hat{A}c+2w^T\hat{c}+\hat{v}^TAc^2 +2\hat{v}^T\hat{A}c+\hat{v}^Tc^3\\&-\frac{4}{3}\hat{w}^Te+\hat{w}^TAc^2-2\hat{w}^TAc+4\hat{w}^Tc+2\hat{w}^T \hat{A}c-2\hat{w}^T\hat{c}+\hat{w}^Tc^3-3\hat{w}^Tc^2=\frac{1}{15}.\\ \end{aligned}$$

Order condition:

$$\begin{aligned} -\frac{1}{30}\theta&+v^TcA^2c+v^TcA\hat{c}+v^Tc\hat{A}c-\frac{2}{3}w^Tc +\frac{1}{6}w^Te+\frac{1}{2}w^Tc^2+w^TcA^2c-w^TA^2c\\&-w^TcAc+w^TAc+w^TcA\hat{c}-w^TA\hat{c}+w^Tc\hat{A}c-w^T\hat{A}c -w^Tc\hat{c}+w^T\hat{c}\\&+\hat{v}^TA^2c+\hat{v}^TA\hat{c}+\hat{v}^T\hat{A}c+\hat{v}^TcAc +\hat{v}^Tc\hat{c}-\frac{2}{3}\hat{w}^Te+2\hat{w}^Tc+\hat{w}^TA^2c\\&-2\hat{w}^TAc+\hat{w}^TA\hat{c}+\hat{w}^T\hat{A}c-2\hat{w}^T\hat{c} +\hat{w}^TcAc-\hat{w}^Tc^2+\hat{w}^Tc\hat{c}=\frac{1}{30}. \end{aligned}$$

Order condition:

$$\begin{aligned} -\frac{1}{20}\theta&+v^T(Ac)^2+2v^T\hat{c}Ac+v^T\hat{c}^2+\frac{1}{4}w^Te +w^TAc+w^T(Ac)^2-w^Tc+w^T\hat{c}\\&-2w^TcAc+2w^T\hat{c}Ac+w^Tc^2-2w^Tc\hat{c}+w^T\hat{c}^2+2\hat{v}^TcAc +2\hat{v}^Tc\hat{c}\\&+3\hat{w}^Tc-\hat{w}^Te+2\hat{w}^TcAc-2\hat{w}^TAc-2\hat{w}^Tc^2 +2\hat{w}^Tc\hat{c}-2\hat{w}^T\hat{c}=\frac{1}{20}. \end{aligned}$$

Order condition:

$$\begin{aligned} -\frac{1}{20}\theta&+v^TAc^3+3v^T\hat{A}c^2+\frac{1}{4}w^Te+w^TAc^3 -3w^TAc^2+3w^TAc-w^Tc\\&+3w^T\hat{A}c^2-6w^T\hat{A}c+3w^T\hat{c}+\hat{v}^Tc^3+\hat{w}^Tc^3 -3\hat{w}^Tc^2+3\hat{w}^Tc-\hat{w}^Te=\frac{1}{20}. \end{aligned}$$

Order condition:

$$\begin{aligned} -\frac{1}{40}\theta&+v^TA(cAc)+v^TA(c\hat{c})+v^T\hat{A}Ac+v^T\hat{A} \hat{c}+v^T\hat{A}c^2+\frac{1}{8}w^Te+\frac{3}{2}w^TAc\\&-\frac{1}{2}w^Tc+w^TA(cAc)-w^TAc^2-w^TA^2c+w^TA(c\hat{c})-w^TA\hat{c} +\frac{3}{2}w^T\hat{c}\\&+w^T\hat{A}Ac-3w^T\hat{A}c+w^T\hat{A}\hat{c}+w^T\hat{A}c^2+\hat{v}^TcAc +\hat{v}^Tc\hat{c}+\frac{3}{2}\hat{w}^Tc-\frac{1}{2}\hat{w}^Te\\&+\hat{w}^TcAc-\hat{w}^TAc-\hat{w}^Tc^2+\hat{w}^Tc\hat{c} -\hat{w}^T\hat{c}=\frac{1}{40}. \end{aligned}$$

Order condition:

$$\begin{aligned} -\frac{1}{60}\theta&+v^TA^2c^2+2v^TA\hat{A}c+v^T\hat{A}c^2-\frac{1}{3}w^Tc +\frac{1}{12}w^Te+w^TA^2c^2-2w^TA^2c\\&+w^TAc+2w^TA\hat{A}c-2w^TA\hat{c}+w^T\hat{A}c^2-2w^T\hat{A}c+w^T\hat{c} +\hat{v}^TAc^2+2\hat{v}^T\hat{A}c\\&-\frac{1}{3}\hat{w}^Te+\hat{w}^TAc^2-2\hat{w}^TAc+\hat{w}^Tc +2\hat{w}^T\hat{A}c-2\hat{w}^T\hat{c}=\frac{1}{60}. \end{aligned}$$

Order condition:

$$\begin{aligned} -\frac{1}{120}\theta&+v^TA^3c+v^TA^2\hat{c}+v^TA\hat{A}c+v^T\hat{A} Ac+v^T\hat{A}\hat{c}+\frac{1}{24}w^Te-\frac{1}{6}w^Tc+\frac{1}{2}w^TAc\\&+w^TA^3c-w^TA^2c+w^TA^2\hat{c}+w^TA\hat{A}c-w^TA\hat{c}+\frac{1}{2}w^T \hat{c}+w^T\hat{A}Ac-w^T\hat{A}c\\&+w^T\hat{A}\hat{c}+\hat{v}^TA^2c+\hat{v}^TA\hat{c}+\hat{v}^T\hat{A}c -\frac{1}{6}\hat{w}^Te+\frac{1}{2}\hat{w}^Tc+\hat{w}^TA^2c-\hat{w}^TAc\\&+\hat{w}^TA\hat{c}+\hat{w}^T\hat{A}c-\hat{w}^T\hat{c}=\frac{1}{120}. \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{6}\theta&+v^Tc^5+w^Tc^5-5w^Tc^4+10w^Tc^3-10w^Tc^2+5w^Tc-w^Te+5\hat{v}^Tc^4\\&+5\hat{w}^Tc^4-20\hat{w}^Tc^3+30\hat{w}^Tc^2-20\hat{w}^Tc+5\hat{w}^Te=\frac{1}{6}.\\ \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{12}\theta&+v^Tc^3Ac+v^Tc^3\hat{c}+\frac{7}{2}w^Tc^3+w^Tc^3 Ac-w^Tc^4+w^Tc^3\hat{c}-\frac{9}{2}w^Tc^2-3w^Tc^2Ac\\&-3w^Tc^2\hat{c}+\frac{5}{2}w^Tc+3w^TcAc+3w^Tc\hat{c}-\frac{1}{2} w^Te-w^TAc-w^T\hat{c}+3\hat{v}^Tc^2Ac\\&+3\hat{v}^Tc^2\hat{c}+\hat{v}^Tc^4+\frac{27}{2}\hat{w}^Tc^2 +3\hat{w}^Tc^2Ac-7\hat{w}^Tc^3+3\hat{w}^Tc^2\hat{c}-10\hat{w}^Tc-6\hat{w}^TcAc\\&-6\hat{w}^Tc\hat{c}+\frac{5}{2}\hat{w}^Te+3\hat{w}^TAc+3 \hat{w}^T\hat{c}+\hat{w}^Tc^4=\frac{1}{12}. \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{30}\theta&+v^TAc^4+4v^T\hat{A}c^4-\frac{1}{5}w^Te+w^TAc^4 -4w^TAc^3+6w^TAc^2-4w^TAc+w^Tc\\&+4w^T\hat{A}c^3-12w^T\hat{A}c^2+12w^T\hat{A}c-4w^T\hat{c} +\hat{v}^Tc^4+\hat{w}^Tc^4-4\hat{w}^Tc^3+6\hat{w}^Tc^2\\&-4\hat{w}^Tc+\hat{w}^Te=\frac{1}{30}. \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{36}\theta&+v^Tc^2A^2c+v^Tc^2A\hat{c}+v^Tc^2\hat{A}c -\frac{7}{6}w^Tc^2+\frac{1}{2}w^Tc^3+w^Tc^2A^2c-w^Tc^2Ac\\&+w^Tc^2A\hat{c}+w^Tc^2\hat{A}c-w^Tc^2\hat{c}+\frac{5}{6}w^Tc -2w^TcA^2c+2w^TcAc-2w^TcA\hat{c}\\&-2w^Tc\hat{A}c+2w^Tc\hat{c}-\frac{1}{6}w^Te+w^TA^2c-w^TAc+w^TA \hat{c}+w^T\hat{A}c-w^T\hat{c}\\&+2\hat{v}^TcA^2c+2\hat{v}^TcA\hat{c}+2\hat{v}^Tc\hat{A}c +\hat{v}^Tc^2Ac+\hat{v}^Tc^2\hat{c}-\frac{10}{3}\hat{w}^Tc +\frac{7}{2}\hat{w}^Tc^2\\&+2\hat{w}^TcA^2c-4\hat{w}^TcAc+2\hat{w}^TcA\hat{c}+2\hat{w}^Tc \hat{A}c-4\hat{w}^Tc\hat{c}+\frac{5}{6}\hat{w}^Te-2\hat{w}^TA^2c\\&+3\hat{w}^TAc-2\hat{w}^TA\hat{c}-2\hat{w}^T\hat{A}c+3\hat{w}^T \hat{c}+\hat{w}^Tc^2Ac-\hat{w}^Tc^3+\hat{w}^Tc^2\hat{c}=\frac{1}{36}. \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{120}\theta&+v^TA^2c^3+3v^TA\hat{A}c^2+v^T\hat{A}c^3 -\frac{1}{20}w^Te+\frac{1}{4}w^Tc+w^TA^2c^3-3w^TA^2c^2\\&+3w^TA^2c-w^TAc+3w^TA\hat{A}c^2-6w^TA\hat{A}c+3w^TA\hat{c} +w^T\hat{A}c^3-3w^T\hat{A}c^2\\&+3w^T\hat{A}c-w^T\hat{c}+\hat{v}^TAc^3+3\hat{v}^T\hat{A}c^2 +\frac{1}{4}\hat{w}^Te+\hat{w}^TAc^3-3\hat{w}^TAc^2+3\hat{w}^TAc\\&-\hat{w}^Tc+3\hat{w}^T\hat{A}c^2-6\hat{w}^T\hat{A}c+3\hat{w}^T\hat{c}=\frac{1}{120}. \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{60}\theta&+v^TA(c^2Ac)+v^TA(c^2\hat{c})+2v^T\hat{A}(cAc) +2v^T\hat{A}(c\hat{c})+v^T\hat{A}c^3-\frac{1}{10}w^Te\\&+\frac{5}{2}w^TAc^2+w^TA(c^2Ac)-w^TAc^3+w^TA(c^2\hat{c})-2w^TAc-2w^TA(cAc)\\&-2w^TA(c\hat{c})+\frac{1}{2}w^Tc+w^TA^2c+w^TA\hat{c}+6w^T\hat{A}c +2w^T\hat{A}(cAc)-5w^T\hat{A}c^2\\&+2w^T\hat{A}(c\hat{c})-2w^T\hat{c}-2w^T\hat{A}Ac-2w^T\hat{A}\hat{c} +w^T\hat{A}c^3+\hat{v}^Tc^2Ac+\hat{v}^Tc^2\hat{c}\\&+\frac{5}{2}\hat{w}^Tc^2+\hat{w}^Tc^2Ac-\hat{w}^Tc^3+\hat{w}^T c^2\hat{c}-2\hat{w}^Tc-2\hat{w}^TcAc-2\hat{w}^Tc\hat{c}+\frac{1}{2}\hat{w}^Te\\&+\hat{w}^TAc+\hat{w}^T\hat{c}=\frac{1}{60}. \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{24}\theta&+v^Tc(Ac)^2+2v^Tc(\hat{c}Ac)+v^Tc\hat{c}^2+\frac{5}{4} w^Tc+3w^TcAc+w^Tc(Ac)^2-2w^Tc^2\\&+3w^Tc\hat{c}-2w^Tc^2Ac+2w^Tc(\hat{c}Ac)+w^Tc^3-2w^Tc^2\hat{c}+w^T c\hat{c}^2-\frac{1}{4}w^Te\\&-w^TAc-w^T(Ac)^2-w^T\hat{c}-2w^T\hat{c}Ac-w^T\hat{c}^2+2\hat{v}^Tc^2 Ac+2\hat{v}^Tc^2\hat{c}\\&+\hat{v}^T(Ac)^2+2\hat{v}^T\hat{c}Ac+\hat{v}^T\hat{c}^2+6\hat{w}^Tc^2 +2\hat{w}^Tc^2Ac-2\hat{w}^Tc^3+2\hat{w}^Tc^2\hat{c}\\&-5\hat{w}^Tc-6\hat{w}^TcAc-6\hat{w}^Tc\hat{c}+\frac{5}{4}\hat{w}^Te+3 \hat{w}^TAc+3\hat{w}^T\hat{c}+\hat{w}^T(Ac)^2\\&+2\hat{w}^T\hat{c}Ac+\hat{w}^T\hat{c}^2=\frac{1}{24}. \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{18}\theta&+v^Tc^2Ac^2+2v^Tc^2\hat{A}c-\frac{7}{3}w^Tc^2+w^Tc^2A c^2-2w^Tc^2Ac+w^Tc^3+2w^Tc^2\hat{A}c\\&-2w^Tc^2\hat{c}+\frac{5}{3}w^Tc-2w^TcAc^2+4w^TcAc-4w^Tc\hat{A}c+4w^T c\hat{c}-\frac{1}{3}w^Te\\&+w^TAc^2-2w^TAc+2w^T\hat{A}c-2w^T\hat{c}+2\hat{v}^TcAc^2+4\hat{v}^Tc \hat{A}c+\hat{v}^Tc^4-\frac{20}{3}\hat{w}^Tc\\&+2\hat{w}^TcAc^2-4\hat{w}^TcAc+8\hat{w}^Tc^2+4\hat{w}^Tc\hat{A}c-4 \hat{w}^Tc\hat{c}+\frac{5}{3}\hat{w}^Te-2\hat{w}^TAc^2\\&+4\hat{w}^TAc-4\hat{w}^T\hat{A}c+4\hat{w}^T\hat{c}+\hat{w}^Tc^4-4 \hat{w}^Tc^3=\frac{1}{18} \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{24}\theta&+v^TcAc^3+3v^Tc\hat{A}c^2+\frac{5}{4}w^Tc+w^TcAc^3-3 w^TcAc^2+3w^TcAc-w^Tc^2\\&+3w^Tc\hat{A}c^2-6w^Tc\hat{A}c+3w^Tc\hat{c}-\frac{1}{4}w^Te-w^TAc^3+3 w^TAc^2-3w^TAc \\&-3w^T\hat{A}c^2+6w^T\hat{A}c-3w^T\hat{c}+\hat{v}^TAc^3+3\hat{v}^T \hat{A}c^2+\hat{v}^Tc^4+\frac{5}{4}\hat{w}^Te+\hat{w}^TAc^3\\&-3\hat{w}^TAc^2+3\hat{w}^TAc-5\hat{w}^Tc+3\hat{w}^T\hat{A}c^2-6 \hat{w}^T\hat{A}c+3\hat{w}^T\hat{c}+\hat{w}^Tc^4-4\hat{w}^Tc^3\\&+6\hat{w}^Tc^2=\frac{1}{24}. \end{aligned}$$

Order condition:

$$\begin{aligned} \begin{aligned} \frac{1}{144}\theta&+v^TcA^3c+v^TcA^2\hat{c}+v^Tc(A\hat{A}c)+v^T c(\hat{A}Ac)+v^Tc\hat{A}\hat{c}+\frac{5}{24}w^Tc-\frac{1}{6}w^Tc^2\\&+\frac{1}{2}w^TcAc+w^TcA^3c-w^TcA^2c+w^TcA^2\hat{c}+w^Tc(A\hat{A}c)-w^T cA\hat{c}+\frac{1}{2}w^Tc\hat{c}\\&+w^Tc(\hat{A}Ac)-w^Tc\hat{A}c+w^Tc\hat{A}\hat{c}-\frac{1}{24}w^Te -\frac{1}{2}w^TAc-w^TA^3c+w^TA^2c\\&-w^TA^2\hat{c}-w^TA\hat{A}c+w^TA\hat{c}-\frac{1}{2}w^T\hat{c}-w^T\hat{A} Ac+w^T\hat{A}c-w^T\hat{A}\hat{c}+\hat{v}^TA^3c\\&+\hat{v}^TA^2\hat{c}+\hat{v}^TA\hat{A}c+\hat{v}^T\hat{A}Ac+\hat{v}^T \hat{A}\hat{c}+\hat{v}^TcA^2c+\hat{v}^TcA\hat{c}+\hat{v}^Tc\hat{A}c +\frac{5}{24}\hat{w}^Te\\&-\frac{5}{6}\hat{w}^Tc+\frac{3}{2}\hat{w}^TAc+\hat{w}^TA^3c-2\hat{w}^T A^2c+\hat{w}^TA^2\hat{c}+\hat{w}^TA\hat{A}c-2\hat{w}^TA\hat{c}\\&+\frac{3}{2}\hat{w}^T\hat{c}+\hat{w}^T\hat{A}Ac-2\hat{w}^T\hat{A}c +\hat{w}^T\hat{A}\hat{c}+\frac{1}{2}\hat{w}^Tc^2+\hat{w}^TcA^2c-\hat{w}^TcAc\\&+\hat{w}^TcA\hat{c}+\hat{w}^Tc\hat{A}c-\hat{w}^Tc\hat{c}=\frac{1}{144}. \end{aligned} \end{aligned}$$

Order condition:

$$\begin{aligned} \begin{aligned} \frac{1}{360}\theta&+v^TA^3c^2+2v^TA^2\hat{A}c+v^TA\hat{A}c^2+v^T\hat{A} Ac^2+2v^T\hat{A}^2c-\frac{1}{60}w^Te+\frac{1}{12}w^Tc\\&-\frac{1}{3}w^TAc+w^TA^3c^2-2w^TA^3c+w^TA^2c+2w^TA^2\hat{A}c-2w^TA^2 \hat{c}+w^TA\hat{A}c^2\\&-2w^TA\hat{A}c+w^TA\hat{c}-\frac{1}{3}w^T\hat{c}+w^T\hat{A}Ac^2-2w^T \hat{A}Ac+w^T\hat{A}c+2w^T\hat{A}^2c\\&-2w^T\hat{A}\hat{c}+\hat{v}^TA^2c^2+2\hat{v}^TA\hat{A}c+\hat{v}^T \hat{A}c^2+\frac{1}{12}\hat{w}^Te-\frac{1}{3}\hat{w}^Tc+\hat{w}^TA^2c^2\\&-2\hat{w}^TA^2c+\hat{w}^TAc+2\hat{w}^TA\hat{A}c-2\hat{w}^TA\hat{c} +\hat{w}^T\hat{A}c^2-2\hat{w}^T\hat{A}c+\hat{w}^T\hat{c}=\frac{1}{360}. \end{aligned} \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{240}\theta&+v^TA^2(cAc)+v^TA^2(c\hat{c})+v^TA\hat{A}Ac +v^TA\hat{A}\hat{c}+v^TA\hat{A}c^2+v^T\hat{A}(cAc)\\&+v^T\hat{A}(c\hat{c})-\frac{1}{40}w^Te+\frac{1}{8}w^Tc+\frac{3}{2} w^TA^2c-\frac{1}{2}w^TAc+w^TA^2(cAc)\\&-w^TA^2c^2-w^TA^3c+w^TA^2(c\hat{c})-w^TA^2\hat{c}+\frac{3}{2}w^TA \hat{c}+w^TA\hat{A}Ac\\&-3w^TA\hat{A}c+w^TA\hat{A}\hat{c}+w^TA\hat{A}c^2+\frac{3}{2}w^T \hat{A}c+w^T\hat{A}(cAc)-w^T\hat{A}c^2\\&+w^T\hat{A}(c\hat{c})-\frac{1}{2}w^T\hat{c}-w^T\hat{A}Ac-w^T \hat{A}\hat{c}+\hat{v}^TA(cAc)+\hat{v}^TA(c\hat{c})\\&+\hat{v}^T\hat{A}Ac+\hat{v}^T\hat{A}\hat{c}+\hat{v}^T\hat{A}c^2 +\frac{1}{8}\hat{w}^Te+\frac{3}{2}\hat{w}^TAc-\frac{1}{2}\hat{w}^Tc+\hat{w}^TA(cAc)\\&-\hat{w}^TAc^2-\hat{w}^TA^2c+\hat{w}^TA(c\hat{c})-\hat{w}^TA\hat{c} +\frac{3}{2}\hat{w}^T\hat{c}+\hat{w}^T\hat{A}Ac\\&-3\hat{w}^T\hat{A}c+\hat{w}^T\hat{A}\hat{c}+\hat{w}^T\hat{A}c^2=\dfrac{1}{240}. \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{120}\theta&+v^TA(Ac)^2+2v^TA(\hat{c}Ac)+v^TA\hat{c}^2+2v^T \hat{A}(cAc)+2v^T\hat{A}(c\hat{c})-\frac{1}{20}w^Te\\&+\frac{1}{4}w^Tc+w^TA(Ac)^2+w^TAc^2+w^TA\hat{c}^2+w^TA^2c-w^T Ac+w^TA\hat{c}\\&-2w^TA(cAc)+2w^TA(\hat{c}Ac)-2w^TA(c\hat{c})+3w^T\hat{A}c+2w^T\hat{A}(cAc)\\&-2w^T\hat{A}c^2+2w^T\hat{A}(c\hat{c})-w^T\hat{c}-2w^T\hat{A}Ac-2w^T \hat{A}\hat{c}+\hat{v}^T(Ac)^2 \\&+2\hat{v}^T\hat{c}Ac+\hat{v}^T\hat{c}^2+\frac{1}{4}\hat{w}^Te+\hat{w}^T (Ac)^2+\hat{w}^Tc^2+\hat{w}^T\hat{c}^2+\hat{w}^TAc\\&-\hat{w}^Tc+\hat{w}^T\hat{c}-2\hat{w}^TcAc+2\hat{w}^T\hat{c}Ac-2 \hat{w}^Tc\hat{c}=\frac{1}{120}. \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{180}\theta&+v^TA(cA^2c)+v^TA(cA\hat{c})+v^TA(c\hat{A}c)+v^T\hat{A} A^2c+v^T\hat{A}A\hat{c}+v^T\hat{A}^2c\\&+v^T\hat{A}(cAc)+v^T\hat{A}(c\hat{c})-\frac{1}{30}w^Te-\frac{2}{3}w^TAc +\frac{1}{2}w^TAc^2+w^TA(cA^2c)\\&-w^TA(cAc)+w^TA(cA\hat{c})+w^TA(c\hat{A}c)-w^TA(c\hat{c})+\frac{1}{6}w^Tc-w^TA^3c\\&+w^TA^2c-w^TA^2\hat{c}-w^TA\hat{A}c+w^TA\hat{c}-\frac{2}{3}w^T\hat{c} +2w^T\hat{A}c+w^T\hat{A}A^2c\\&-2w^T\hat{A}Ac+w^T\hat{A}A\hat{c}+w^T\hat{A}^2c-2w^T\hat{A}\hat{c}+w^T \hat{A}(cAc)-w^T\hat{A}c^2\\&+w^T\hat{A}(c\hat{c})+\hat{v}^TcA^2c+\hat{v}^TcA\hat{c}+\hat{v}^Tc\hat{A}c -\frac{2}{3}\hat{w}^Tc+\frac{1}{2}\hat{w}^Tc^2+\hat{w}^TcA^2c\\&-\hat{w}^TcAc+\hat{w}^TcA\hat{c}+\hat{w}^Tc\hat{A}c-\hat{w}^Tc\hat{c} +\frac{1}{6}\hat{w}^Te-\hat{w}^TA^2c+\hat{w}^TAc\\&-\hat{w}^TA\hat{c}-\hat{w}^T\hat{A}c+\hat{w}^T\hat{c}=\frac{1}{180}. \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{72}\theta&+v^T(Ac)(A^2c)+v^T(Ac)(A\hat{c})+v^T(Ac)(\hat{A}c) +v^T\hat{c}A^2c+v^T\hat{c}A\hat{c}+v^T\hat{c}\hat{A}c\\&-\frac{1}{12}w^Te+\frac{5}{12}w^Tc+\frac{1}{2}w^TA^2c-\frac{2}{3}w^TAc +\frac{1}{2}w^TA\hat{c}+\frac{1}{2}w^T\hat{A}c-\frac{2}{3}w^T\hat{c}\\&+\frac{3}{2}w^TcAc+w^T(Ac)(A^2c)-w^T(Ac)^2+w^T(Ac)(A\hat{c})+w^T(Ac)(\hat{A}c)\\&-2w^T\hat{c}Ac-\frac{1}{2}w^Tc^2-w^TcA^2c-w^TcA\hat{c}-w^Tc\hat{A}c +\frac{3}{2}w^Tc\hat{c}+w^T\hat{c}A^2c\\&+w^T\hat{c}A\hat{c}+w^T\hat{c}\hat{A}c-w^T\hat{c}^2+\hat{v}^TcA^2c +\hat{v}^TcA\hat{c}+\hat{v}^Tc\hat{A}c+\hat{v}^T(Ac)^2\\&+2\hat{v}^T\hat{c}Ac+\hat{v}^T\hat{c}^2-\frac{5}{3}\hat{w}^Tc +\frac{3}{2}\hat{w}^Tc^2+\hat{w}^TcA^2c-3\hat{w}^TcAc+\hat{w}^TcA\hat{c}\\&+\hat{w}^Tc\hat{A}c-3\hat{w}^Tc\hat{c}+\frac{5}{12}\hat{w}^Te -\hat{w}^TA^2c+2\hat{w}^TAc-\hat{w}^TA\hat{c}-\hat{w}^T\hat{A}c+2\hat{w}^T\hat{c}\\&+\hat{w}^T(Ac)^2+\hat{w}^T\hat{c}^2+2\hat{w}^T\hat{c}Ac=\frac{1}{72}. \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{48}\theta&+v^TcA(cAc)+v^TcA(c\hat{c})+v^Tc(\hat{A}Ac)+v^T c\hat{A}\hat{c}+v^Tc\hat{A}c^2+\frac{5}{8}w^Tc+\frac{3}{2}w^TcAc\\&-\frac{1}{2}w^Tc^2+w^TcA(cAc)-w^TcAc^2-w^TcA^2c+w^TcA(c\hat{c})-w^TcA\hat{c} +\frac{3}{2}w^Tc\hat{c}\\&+w^Tc(\hat{A}Ac)-3w^Tc\hat{A}c+w^Tc\hat{A}\hat{c}+w^Tc\hat{A}c^2 -\frac{1}{8}w^Te-\frac{3}{2}w^TAc-w^TA(cAc)\\&+w^TAc^2+w^TA^2c-w^TA(c\hat{c})+w^TA\hat{c}-\frac{3}{2}w^T\hat{c} -w^T\hat{A}Ac+3w^T\hat{A}c-w^T\hat{A}\hat{c}\\&-w^T\hat{A}c^2+\hat{v}^TA(cAc)+\hat{v}^TA(c\hat{c})+\hat{v}^T\hat{A}A c+\hat{v}^T\hat{A}\hat{c}+\hat{v}^T\hat{A}c^2+\hat{v}^Tc^2Ac\\&+\hat{v}^Tc^2\hat{c}+\frac{5}{8}\hat{w}^Te+\frac{5}{2}\hat{w}^TAc -\frac{5}{2}\hat{w}^Tc+\hat{w}^TA(cAc)-\hat{w}^TAc^2-\hat{w}^TA^2c\\&+\hat{w}^TA(c\hat{c})-\hat{w}^TA\hat{c}+\frac{5}{2}\hat{w}^T\hat{c} +\hat{w}^T\hat{A}Ac-3\hat{w}^T\hat{A}c+\hat{w}^T\hat{A}\hat{c} +\hat{w}^T\hat{A}c^2\\&+\frac{5}{2}\hat{w}^Tc^2+\hat{w}^Tc^2Ac-\hat{w}^Tc^3+\hat{w}^Tc^2 \hat{c}-2\hat{w}^TcAc-2\hat{w}^Tc\hat{c}=\frac{1}{48}. \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{90}\theta&+v^TA(cAc^2)+2v^TA(c\hat{A}c)+v^T\hat{A}Ac^2+2v^T \hat{A}^2c+v^T\hat{A}c^3-\frac{1}{15}w^Te\\&-\frac{4}{3}w^TAc+w^TA(cAc^2)-2w^TA(cAc)+w^TAc^2+2w^TA(c\hat{A}c)-2 w^TA(c\hat{c})\\&+\frac{1}{3}w^Tc-w^TA^2c^2+2w^TA^2c-2w^TA\hat{A}c+2w^TA\hat{c} -\frac{4}{3}w^T\hat{c}+w^T\hat{A}Ac^2\\&-2w^T\hat{A}Ac+4w^T\hat{A}c+2w^T\hat{A}^2c-2w^T\hat{A}\hat{c}+w^T \hat{A}c^3-3w^T\hat{A}c^2+\hat{v}^TcAc^2\\&+2\hat{v}^Tc\hat{A}c-\frac{4}{3}\hat{w}^Tc+\hat{w}^TcAc^2-2\hat{w}^TcAc +\hat{w}^Tc^2+2\hat{w}^Tc\hat{A}c-2\hat{w}^Tc\hat{c}\\&+\frac{1}{3}\hat{w}^Te-\hat{w}^TAc^2+2\hat{w}^TAc-2\hat{w}^T\hat{A}c+2 \hat{w}^T\hat{c}=\frac{1}{90}. \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{72}\theta&+v^TcA^2c^2+2v^Tc(A\hat{A}c)+v^Tc\hat{A}c^2 +\frac{5}{12}w^Tc-\frac{1}{3}w^Tc^2+w^TcA^2c^2\\&-2w^TcA^2c+w^TcAc+2w^Tc(A\hat{A}c)-2w^TcA\hat{c}+w^Tc\hat{A}c^2-2w^Tc\hat{A}c\\&+w^Tc\hat{c}-\frac{1}{12}w^Te-w^TA^2c^2+2w^TA^2c-w^TAc-2w^T A\hat{A}c+2w^TA\hat{c}\\&-w^T\hat{A}c^2+2w^T\hat{A}c-w^T\hat{c}+\hat{v}^TA^2c^2 +2\hat{v}^TA\hat{A}c+\hat{v}^T\hat{A}c^2+\hat{v}^TcAc^2\\&+2\hat{v}^Tc\hat{A}c+\frac{5}{12}\hat{w}^Te-\frac{5}{3} \hat{w}^Tc+\hat{w}^TA^2c^2-2\hat{w}^TA^2c+3\hat{w}^TAc\\&+2\hat{w}^TA\hat{A}c-2\hat{w}^TA\hat{c}+\hat{w}^T\hat{A} c^2-4\hat{w}^T\hat{A}c+3\hat{w}^T\hat{c}+\hat{w}^TcAc^2\\&-2\hat{w}^TcAc+\hat{w}^Tc^2+2\hat{w}^Tc\hat{A}c-2\hat{w}^T c\hat{c}-\hat{w}^TAc^2=\frac{1}{72}. \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{36}\theta&+v^T(Ac)(Ac^2)+2v^T(Ac)(\hat{A}c)+v^T\hat{c} Ac^2+2v^T\hat{c}\hat{A}c-\frac{1}{6}w^Te+\frac{1}{2}w^TAc^2\\&-\frac{4}{3}w^TAc+\frac{5}{6}w^Tc+w^T\hat{A}c-\frac{4}{3}w^T \hat{c}+w^T(Ac)(Ac^2)-2w^T(Ac)^2\\&+3w^TcAc+2w^T(Ac)(\hat{A}c)-4w^T\hat{c}Ac-w^TcAc^2-w^Tc^2-2 w^Tc\hat{A}c\\&+3w^Tc\hat{c}+w^T\hat{c}Ac^2+2w^T\hat{c}\hat{A}c-2w^T\hat{c}^2 +\hat{v}^TcAc^2+2\hat{v}^Tc\hat{A}c+\hat{v}^Tc^2Ac\\&+\hat{v}^Tc^2\hat{c}-\frac{10}{3}\hat{w}^Tc+\hat{w}^TcAc^2-4 \hat{w}^TcAc+\frac{7}{2}\hat{w}^Tc^2+2\hat{w}^Tc\hat{A}c-4\hat{w}^Tc\hat{c}\\&+\frac{5}{6}\hat{w}^Te-\hat{w}^TAc^2+3\hat{w}^TAc-2\hat{w}^T \hat{A}c+3\hat{w}^T\hat{c}+\hat{w}^Tc^2Ac-\hat{w}^Tc^3\\&+\hat{w}^Tc^2\hat{c}=\frac{1}{36}. \end{aligned}$$

Order condition:

$$\begin{aligned} \frac{1}{720}\theta&+v^TA^4c+v^TA^3\hat{c}+v^TA^2\hat{A}c+v^TA \hat{A}Ac+v^TA\hat{A}\hat{c}+v^T\hat{A}A^2c\\&+v^T\hat{A}A\hat{c}+v^T\hat{A}^2c-\frac{1}{120}w^Te+\frac{1}{24}w^T c-\frac{1}{6}w^TAc+\frac{1}{2}w^TA^2c\\&+w^TA^4c-w^TA^3c+w^TA^3\hat{c}+w^TA^2\hat{A}c-w^TA^2\hat{c} +\frac{1}{2}w^TA\hat{c}\\&+w^TA\hat{A}Ac-w^TA\hat{A}c+w^TA\hat{A}\hat{c}-\frac{1}{6}w^T \hat{c}+\frac{1}{2}w^T\hat{A}c+w^T\hat{A}A^2c\\&-w^T\hat{A}Ac+w^T\hat{A}A\hat{c}+w^T\hat{A}^2c-w^T\hat{A}\hat{c} +\hat{v}^TA^3c+\hat{v}^TA^2\hat{c}+\hat{v}^TA\hat{A}c\\&+\hat{v}^T\hat{A}Ac+\hat{v}^T\hat{A}\hat{c}+\frac{1}{24}\hat{w}^Te -\frac{1}{6}\hat{w}^Tc+\frac{1}{2}\hat{w}^TAc+\hat{w}^TA^3c-\hat{w}^TA^2c\\&+\hat{w}^TA^2\hat{c}+\hat{w}^TA\hat{A}c-\hat{w}^TA\hat{c} +\frac{1}{2}\hat{w}^T\hat{c}+\hat{w}^T\hat{A}Ac-\hat{w}^T\hat{A}c+\hat{w}^T \hat{A}\hat{c}=\frac{1}{720}. \end{aligned}$$

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Mukaddes Ökten Turaci
- 1
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Turgut Öziş
- 2

1.Department of Computer Programming, Yenice Vocational SchoolKarabük UniversityKarabükTurkey
2.Department of MathematicsEge UniversityİzmirTurkey

On explicit two-derivative two-step Runge–Kutta methods

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