Stability Optimization of Explicit Runge–Kutta Methods with Higher-Order Derivatives

The paper is devoted to the parametric stability optimization of explicit Runge–Kutta methods with higher-order derivatives. The key feature of these methods is the dependence of the coefficients of their stability polynomials on free parameters. Thus, the integral characteristics of stability domains can be considered as functions of free parameters. The optimization is based on the numerical maximization of the area of the stability domain and the length of the stability interval. Runge–Kutta methods with higher-order derivatives, presented in previous works, are optimized. The optimal values of parameters are computed for methods of fourth, fifth, and sixth orders. In numerical experiments, optimal parameter values are used for the construction of high-order schemes for the method of lines for problems with partial differential equations. Problems for linear and nonlinear hyperbolic and parabolic equations are considered. Additionally, an optimized scheme is used in lattice Boltzmann simulations of gas flow. As the main result of computations and comparison with existing methods, it is demonstrated that optimized schemes have better stability properties and can be used in practice.