# Stability Optimization of Explicit Runge-Kutta Methods with Higher-Order Derivatives
## 6 Conclusions  
In the present study, the stability optimization of RKMs with higher-order derivatives is realized. The approach for optimization is based on the structure of the stability polynomials of methods--their coefficients depend on free parameters. Thus, the geometry of the stability domain can be modified by variations of the parameter values. This idea is close to the approach proposed in [16]. The numerical optimization is realized by the maximization of the area of the stability domain and the length of the stability interval. For the discussion, the following notions can be made:  
1. The optimization of methods proposed in [28; 31; 33] is realized. Optimal values of parameters are computed for methods of fourth, fifth, and sixth orders. Obtained values are  
Figure 12: Plots of the logarithm of the practical estimations of local errors for GJRKMs. Figure 13: Work-precision plots for fourth-order GJRKM, applied to the problem for the system of kinetic equations. used in numerical examples, based on the application of MOL to the solution of problems for linear and nonlinear PDEs. The results of computations demonstrate that, by taking into account the behavior of the spectrum of the Jacobian matrix, better stability properties are demonstrated for schemes based on RKMs with optimal parameters;  
2. As can be seen from Tables 6-9, the largest values of CFL number in the case of constant time step are observed for optimized methods. According to the expressions \(c\Delta t/h\) and \(D\Delta t/h^{2}\) for hyperbolic and parabolic CFL numbers, optimized methods provide the possibility to perform computations with a larger time step, but with the same accuracy order as the non-optimized methods. This leads to a decrease in the total number of time steps and, as a consequence, a decrease in computational time. In the case of the variable time step, as can be seen from Figures 10 and 13, the use of optimized methods leads to a decrease in the right part of evaluations and, as a consequence, to a decrease in computational time;  
3. The obtained values of free parameters are independent of the approach to spatial discretization used in MOL. Moreover, they can be used for the construction of computational schemes based on the modified MOL, as proposed in [42];  
4. The limitation of the proposed approach relates to the fixed number of stages of the considered method and, as a consequence, to the bounded number of free parameters. For strongly stiff problems, the constructed methods are not applicable as explicit methods in comparison to the explicit stabilized RKC method, which has low order, but their stability intervals can be enlarged by an increase in stage number;  
5. For the optimization, we consider two 'opposite' cases: when stability domain is maximized on its area and its length on the real negative axis. In future studies, this problem can be stated as a multicriterial problem, or a problem of the optimization of composite functional, which is constructed as the convex combination \(\zeta S(\mathbf{q})+(1-\zeta)\mathcal{L}(\mathbf{q})\), where \(\zeta\in[0,1]\) (similar approaches are used in [39; 48]). In the present paper, we try to estimate the perspectives of the optimization of high-order RKMs. Other optimization problems can be solved in future studies;  
6. Another associated problem with the optimization of RKMs is the problem of the simultaneous optimization of stability and accuracy properties by free parameters. For hyperbolic PDEs, this problem is associated with the construction of low-dispersion and low-dissipation RKMs, where the stability and accuracy characteristics are simultaneously optimized. This problem is considered in many works (see [39; 48; 49; 50; 51; 52]). The optimization of TDRKM in the case of \(s=4\) is realized in [39], where it is demonstrated that the accuracy properties and performance of the optimized method are better than for other well-known methods.