# Construction of Two-Derivative Runge-Kutta Methods of Order Six
## 1 Introduction  
We consider systems of first-order ODEs of the form  
\[y^{\prime}(x)=f(x,y(x)),\quad x\in[x_{0},X],\quad y(x_{0})=y_{0}. \tag{1}\]  
These problems can be solved efficiently using Runge-Kutta (RK) or multi-step methods. Numerical methods of a high algebraic order are efficient integrators. The computational cost increases with the order, for example, for the RK methods, it is well known that seven, nine, and eleven stages are needed for orders six, seven, and eight, respectively. To reduce the computational cost, research has been devoted to methods with special properties suitable for specific problems. Special methods for the integration of problems with oscillatory behavior of the solution have been considered by several authors from the early stages of research on ODE solvers. There are two general classes of numerical methods for the integration of oscillatory problems. One class consists of methods with frequency-dependent coefficients, while the other includes constant coefficients. In the first class are exponentially, trigonometrically, or phase-fitted methods (see [1, 2, 3]); for these methods, a good estimate of the frequency of the specific problem is needed. The advantage of these methods in the second general class is that they can be applied to every oscillatory problem since the coefficients are constant. Among these are methods with low dissipation and low dispersion, (see [4, 5, 6]). The dispersion (or phase-lag) property was introduced in the pioneer paper of [PERSON] and [PERSON] [7]. In this work, we consider Two-Derivative RK methods with special properties and constant coefficients. These methods originate in the work of [PERSON] and [PERSON] [8, 9], of which they introduced methods where the values of the derivatives of \(f(x,y(x))\), withrespect to \(x\), as well as the values of the function \(f(x,y)\) at some intermediate points, are used. [PERSON] and [PERSON] [10] considered the case where only the first derivative of \(f(x,y(x))\), with respect to \(x\left(g(x,y(x))\,:\,=\,y^{\prime\prime}\,=\,f^{\prime}f\right)\), as well as the function \(f(x,y)\) are evaluated at each step of the method. They call these methods Two-Derivative Runge-Kutta (TDRK) methods. The TDRK method are of the form  
\[Y_{i} = y_{n}+h\sum_{j=1}^{s}a_{ij}f(x_{n}+c_{j}h,Y_{j})+h^{2}\sum_{j=1}^ {s}\hat{a}_{ij}g(x_{n}+c_{j}h,Y_{j})\] \[y_{n+1} = y_{n}+h\sum_{i=1}^{s}b_{i}f(x_{n}+c_{i}h,Y_{i})+h^{2}\sum_{i=1}^ {s}\hat{b}_{i}g(x_{n}+c_{i}h,Y_{i})\]  
the associated Butcher tableau is  
\[ \begin{array}{c|c||c}c&A&\hat{A}\\ \hline&b^{T}&\hat{b}^{T}\end{array} \]  
where \(A\) and \(\hat{A}\) are \(s\times s\) matrices, and \(b\) and \(\hat{b}\), \(c\) are \(s\times 1\) vectors. [PERSON] and [PERSON] derived order conditions for the TDRK methods based on [PERSON]'s algebraic theory of trees [11] in a different way than the conditions were derived in [8; 9]. They considered explicit TDRK methods and gave algebraic order conditions up to order five. Specifically in [10], the special class of explicit TDRK (ETDRK) methods was introduced, where only the first derivative of \(f(x,y(x))\), with respect to x, is evaluated at intermediate points at each step. In this case, the matrix \(A\) has non-zero elements only in the first column i.e., the function \(f\) is evaluated only at \(x_{n}\); these methods are called special ETDRK methods. We shall refer to methods where non-zero entries are allowed into the matrix \(A\) at any column as general ETDRK methods. In [10], conditions are given up to order seven for special ETDRK methods. Since then, research has focused on special ETDRK methods only, where several authors have constructed minimum phase-lag, trigonometrically fitted, phase-fitted methods (see [12; 13; 14]). The authors were the first to consider the general case; in [15], they presented methods of an algebraic order up to five, and in [14], they derived order conditions for trigonometrically fitted methods.