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

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.

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:

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

Order condition:

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

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:

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