# Stability Optimization of Explicit Runge-Kutta Methods with Higher-Order Derivatives
## 4 Maximization of the Integral Characteristics of RKMs with Higher-Order Derivatives  
As can be seen, the stability polynomials in (7)-(11) have terms that are dependent on the input parameters. Thus, the stability polynomials are represented as \(\mathcal{R}=\mathcal{R}(z,\mathbf{q})\), where \(\mathbf{q}\in R^{n}\) is a vector of input parameters. According to this dependency, the geometric form of the stability domain \(\mathcal{D}\) is dependent on the values of \(\mathbf{q}\). We may characterize this domain by two integral characteristics: by its area \(S(\mathbf{q})\) and by the length of the stability interval \(\mathcal{L}(\mathbf{q})\), which is defined as the length of the stability domain on the real negative axis:  
\[S(\mathbf{q})=\iint\limits_{\mathcal{D}(\mathbf{q})}dxdy,\quad\mathcal{L}( \mathbf{q})=-\int\limits_{\mathcal{L}(\mathbf{q})}^{0}dx. \tag{15}\]  
The following computational method for stability optimization can be proposed. Let us know the domain where the values of \(\mathbf{q}\) are defined: \(\mathbf{q}\in Q\subset R^{n}\). For the simplification of the computations, let us propose that \(Q\) is a rectangle in \(R^{n}\): \(Q=\{q_{i},i=1,\ldots,n|q_{i}\in[\underline{q}_{i},\overline{q}^{i}]\}\), where boundary values \(\underline{q}_{i}\) and \(\overline{q}^{i}\) are considered as known. It must be noted that these values can be obtained by a computational analysis of the domain \(Q(\mathbf{q})\) by varying \(\mathbf{q}\) or by a theoretical analysis of terms in the expression for \(\mathcal{R}(z,\mathbf{q})\), which are dependent on \(\mathbf{q}\). The first approach leads to the procedure of the localization of parameter values in some intervals, and the second one, for example, can provide the possibility to exclude the large values of some parameters if they are presented in denominators.  
Let \(\mathbf{q}_{S}\) be the optimal value of \(\mathbf{q}\), which is obtained as a solution of the following optimization problem: \(S(\mathbf{q})\rightarrow\underset{\mathbf{q}\in Q}{\text{max}}\), and \(\mathbf{q}_{L}\) is obtained by the maximization of \(\mathcal{L}(\mathbf{q})\): \(\mathcal{L}(\mathbf{q})\rightarrow\underset{\mathbf{q}\in Q}{\text{max}}\). Optimal parameters, obtained after the maximization of \(S\), can be used in situations when the Jacobian matrix has a full complex spectrum. Parameters that maximize \(\mathcal{L}\) can be used when a matrix has a real spectrum or when the imaginary parts of eigenvalues are situated close to the real negative axis.  
According to the unknown analytical dependencies of \(S\) and \(\mathcal{L}\) on \(\mathbf{q}\), the proposed approach can only be realized computationally because the integrals in Formula (15) can only be computed numerically. For this computation, the rectangular quadrature is used.  
The maximization problem is solved by the semi-heuristic approach, based on two stages. According to the absence of exact expressions for derivatives, we should use only methods that are based on the values of the objective functions and do not use their derivatives. We realize the solution of the maximization problems by two steps. First, we use a random search in \(Q\) for the calculation of the approximate initial values of optimal parameters. In computation, we use \(5\cdot 10^{6}\) uniformly distributed random points. The generation of random numbers is performed by the function rand() in Matlab (version R2021a). At the second stage, we apply the direction set method [44], based on the transition to the solution of 1D problems along every coordinate direction. These problems are solved with the golden section search [44], with accuracy equal to \(10^{-8}\).  
It should be noted that the stated maximization problems can have multiple solutions. In the present study, we do not investigate the uniqueness of the solution and sensitivity to parameter variations. Numerical solutions presented in this paper can be considered as possible optimal solutions.