# Stability Optimization of Explicit Runge-Kutta Methods with Higher-Order Derivatives
## 2 Explicit RKMs with Higher Order Derivatives
### Two Derivative RKMs (TDRKMs)  
In [33], [PERSON] and [PERSON] proposed an explicit method, based on the second derivative of \(y(t)\), which is presented as \(y^{\prime\prime}(t)=g(y)=f^{{}^{\prime}}(y)f(y)\). The explicit method with \(s\) stages is presented as  
\[y_{n+1}=y_{n}+\Delta tf(y_{n})+\Delta t^{2}\sum\limits_{i=1}^{s}\widehat{b}_{ i}g(Y_{i}), \tag{3}\]  
\[Y_{i}=y_{n}+\Delta tc_{i}f(y_{n})+\Delta t^{2}\sum\limits_{j=1}^{s}\widehat{a }_{i,j}g(Y_{j}), \tag{4}\]  
where \(c_{1}=0\), \(\widehat{a}_{i,j}=0\), \(i\leq j\), and \(\sum\limits_{j=1}^{s}\widehat{a}_{i,j}=c_{i}^{2}/2\). Parameters \(c_{i}\) can be treated as the nodes of the method.  
As is demonstrated in [33], unique second- and fourth-order methods with one and two stages, respectively, are constructed. For \(s=3\) and \(s=4\), fifth- and sixth-order methods with free parameters are constructed.  
In the case of \(s=3\), five order conditions on six unknowns \(c_{2}\), \(c_{3}\), \(\widehat{a}_{3,2}\), \(\widehat{b}_{1}\), \(\widehat{b}_{2}\), \(\widehat{b}_{3}\) are considered. The authors of [33] on p. 180 presented expressions of the parameters as functions of the input parameter \(c_{3}\). The following restrictions are imposed on \(c_{3}\)[33]: \(c_{3}\
eq 0\), \(c_{3}\
eq 3/5\), \(c_{3}\
eq 1/2\).  
In the case of \(s=4\), eight order conditions and ten unknowns \(c_{2}\), \(c_{3}\), \(c_{4}\), \(\widehat{a}_{3,2}\), \(\widehat{a}_{4,2}\), \(\widehat{a}_{4,3}\), \(\widehat{b}_{1}\), \(\widehat{b}_{2}\), \(\widehat{b}_{3}\), \(\widehat{b}_{4}\) are considered. On p. 181 from [33], expressions for \(\widehat{b}_{1}\), \(\widehat{b}_{2}\), \(\widehat{b}_{3}\), \(\widehat{b}_{4}\), \(\widehat{a}_{3,2}\), \(\widehat{a}_{4,2}\), and \(\widehat{a}_{4,3}\) as functions of \(c_{2}\), \(c_{3}\), and \(c_{4}\) are presented. So, the three following pairs of the input parameters are considered: \((c_{2},c_{3})\), \((c_{3},c_{4})\), or \((c_{2},c_{4})\). If we set the values of parameters from one of these pairs, the value of another parameter (\(c_{4}\), \(c_{2}\), or \(c_{3}\), respectively) can be obtained from the following order condition (p. 181, [33]):  
\[\bigg{(}\frac{1}{6}c_{3}c_{4}-\frac{1}{12}(c_{3}+c_{4})+\frac{1}{20}\bigg{)}c _{2}^{3}(c_{4}-c_{3})-\bigg{(}\frac{1}{6}c_{2}c_{4}-\frac{1}{12}(c_{2}+c_{4}) +\frac{1}{20}\bigg{)}c_{3}^{3}(c_{4}-c_{2})+\]  
\[+\bigg{(}\frac{1}{6}c_{2}c_{3}-\frac{1}{12}(c_{2}+c_{3})+\frac{1}{20}\bigg{)}c _{4}^{3}(c_{3}-c_{2})=\frac{1}{30}(c_{3}-c_{2})(c_{4}-c_{2})(c_{4}-c_{3}).\]  
This equation has three roots--two of them are equal to the values of the input parameters and the other is represented as a function of these parameters. For example, when \(c_{2}\) and \(c_{4}\) are considered as input parameters, this cubic equation on \(c_{3}\) has the three following roots: \(c_{3}=c_{2}\), \(c_{3}=c_{4}\), and  
\[c_{3}=\frac{-3c_{2}+5c_{2}c_{4}+2-3c_{4}}{10c_{2}c_{4}-5c_{2}+3-5c_{4}}.\]  
For all three variants, the values of input parameters should satisfy the following restrictions: \(c_{2}\
eq 0\), \(c_{3}\
eq 0\), \(c_{4}\
eq 0\), \(c_{2}\
eq c_{3}\
eq c_{4}\), and \(10c_{2}c_{4}-5(c_{2}