# Stability Optimization of Explicit Runge-Kutta Methods with Higher-Order Derivatives
## 1 Introduction
After that, the coefficients of the method are obtained. Fourth-order methods with 12-14 stages are constructed and applied to the solution of Maxwell equations. [PERSON] and [PERSON], in [16], proposed an optimal method with respect to large stability intervals on the imaginary axis (unlike the previously mentioned approaches, where the interval on the real negative axis is maximized). It must be noted that, in the proposed approach, the parameters of the method are used for optimization. The coefficients of stability polynomials depend on these parameters. According to this approach, optimal parameters are obtained through the construction of optimal polynomials, and their values can be substituted into the computational scheme of the method. As such, in this approach, we do not need to perform the construction of method. In [17; 18; 19; 20], the so-called paired RKMs are constructed. The stability polynomials of such RKMs depend on the free parameters, which are chosen in order to perform computations with the largest time step and appropriate stability properties. These methods have many parameters, which are equal to zero, so they are considered quite effective from a computational point of view. Authors present many examples of the applications of such methods in computational fluid dynamics. The approach to the construction of optimal stability polynomials for the explicit method, oriented toward the application to problems for hyperbolic equations and systems, is proposed in [21]. In [22], the authors proposed an original approach to stabilization, based on the use of the Lyapunov function for a system of solved ODEs. In [23], the authors proposed a technique to stabilize standard explicit RKMs by associating them with W-methods, used for the solution of stiff problems. As has been demonstrated, for any given explicit four-stage fourth-order RKM, two uniparametric families of third-order W-methods can be obtained. The free parameter can be used to increase the stability properties of the W-methods. Methods, based on the application of time-accurate and highly stable explicit operators, are proposed in [24; 25; 26]. The methods are developed for application to stiff problems and proposed operators act as preconditioners on the stiff terms and can be deployed to any existing explicit time-marching methods straightforwardly. In the last decades, RKMs with higher-order derivatives have been considered in many works. The higher derivative terms in RKMs give more freedom and provide the possibility to construct methods with higher orders. These derivatives are used for the evaluation of a number of terms in the Taylor expansion for local error [27]. An advantage of these methods over classical RKMs is that they can reach a higher order with fewer function evaluations. Although the computations of higher-order derivative terms put some restrictions on these methods, the advantage gained may still be beneficial [28]. The review of multi-derivative methods is presented in [29]. [PERSON] and [PERSON] in [30] construct explicit and semi-implicit methods up to orders 7 and 6, respectively. [PERSON] and [PERSON] in [31] proposed methods with the computation of derivatives in the internal stages of the method. Extended RKMs with the use of derivatives at external stages are proposed in [32]. [PERSON] et al. [33] proposed two-derivative RKMs (TDRKMs), where the second derivative is incorporated in both internal and external stages. [PERSON] et al. [34] used the trigonometrical fitting to TDRKMs for solving oscillatory ODEs. In [35], TDRKMs are extended for the solution of the Schrodinger equation. [PERSON] et al. [36] proposed the exponentially fitted TDRKM of Nystrom type. [PERSON] and [PERSON] [37] constructed sixth-order TDRKMs. In [38; 39], the authors presented optimized TDRKMs for the simulation of oscillating and wave processes. Strong-stability TDRKMs for application to the numerical solutions of PDEs are proposed in [40]. [PERSON] et al.