# Stability Optimization of Explicit Runge-Kutta Methods with Higher-Order Derivatives
## 1 Introduction  
Runge-Kutta methods (RKMs) have been used in many works for solving the Cauchy problem for ordinary differential equations (ODEs). Many computational schemes for the solution of problems with partial differential equations (PDEs) are based on the method of lines (MOL), based on an approach wherein spatial derivatives are discretized in order to transform PDEs into a semi-discrete system of ODEs. For practical computations, the time step size should be taken as large as possible to yield good computational performance since the cost of solving up to a fixed time is proportional to the number of time steps. For many problems that arise in electrodynamics and fluid mechanics, the number of ODEs can easily reach \(10^{8}\) or higher [1], so explicit methods are preferable. But explicit schemes are conditionally stable, and the used time step is limited. The problem of obtaining optimal stability polynomials for the explicit methods has been studied by many authors [2; 3]. The widely used approach to optimization has two stages. At first, a stability polynomial of fixed degree is constructed. In many works, this is realized in such a way that stability domain or stability interval should be as large as possible. In most papers, this approach is based on the use of Chebyshev polynomials [4]. At the second stage, an explicit method corresponding to the obtained polynomial is constructed. Usually, such methods have a low order, and additional stages are exploited to increase the stability region. Lebedev-type methods [5] are based on optimal-stability polynomials, but an appropriate ordering of zeros is needed for ensuring stability properties. In [6], an approach based on a linear combination of shifted Chebyshev polynomials is proposed. It is based on the use of a three-term recurrence relation. But the stability region on the negative real axis is only about 80% of the optimal interval. In [7], the combined approach, based on the use of orthogonal polynomials of second order, is proposed. [PERSON], in [8],proposed an analytical approach to the construction of an optimal-stability polynomial. In [9], the problem of stability polynomial construction is expressed in terms of the general least-deviation feasibility problem. Its solution is obtained by an algorithm based on convex optimization techniques. [PERSON] and [PERSON], in [10], proposed a special two-step form of RKM for application to PDEs. Their method simplifies the choice of temporal step sizes, satisfying the von Neumann stability criterion by embedding a thin and very long rectangle inside the stability region. [PERSON] and [PERSON], in [11], considered the method for ODEs with Jacobian matrix spectra in the shape of thin regions, placed along the real axis, but also embedded into the whole complex plane. In [12], a procedure for the construction of algorithms of explicit Runge-Kutta-Chebyshev (RKC) methods for stiff problems is proposed, and second-order methods with up to 320 stages are derived. In [13], approaches to the construction of stabilized explicit methods of higher orders are considered. In [14], extrapolated stabilized methods based on Chebyshev polynomials are proposed. [PERSON] and [PERSON] [15] proposed mixed-precision methods, based on RKC schemes. Three mixed-precision schemes (a first- and a second-order RKC method and a first-order multirate RKC scheme) are constructed. These schemes perform only the few function evaluations needed for accuracy (1 or 2 for first- and second-order methods, respectively) in high precision, while the rest are performed in low precision. Original approaches to optimization without the application of Chebyshev polynomials are described in [16; 17; 18; 19; 20; 1]. [PERSON] et al. [1] proposed a general approach to obtain low-storage RKMs with enlarged and tailored stability domains. The construction of stability polynomials is based on the fixation of the desired shape of the stability regionwith the spectra of ODEs derived by MOL from an original PDE.