Optimization of calculations in methods of the Rosenbrock type

Abstract

The paper presents a class of L-stable methods of the third and fourth orders of accuracy m=3 and 4, built on the basis of methods of the Rosenbrock type, in which it is not necessary to calculate the m right-hand parts of the system of differential equations. At each step of integration, they require only two calls to the right-hand sides of the system of differential equations and satisfy the conditions of both absolute and Johr stability. The implementation of the given methods is as simple as Rosenbrock's methods, but the given schemes have better properties of accuracy and stability. As for implicit methods of the Runge-Kutta type, for them, the computational costs are highly dependent on the implementation method. In the proposed methods, one calculation of the Jacobi matrix at the integration step and its LU-factorization is sufficient. The following calculation of the coefficients k_i of the linear combination requires the application of only two procedures of the Gaussian inversion. When integrating the ZDR system with a constant step, it is advisable to "freeze" the Jacobi matrix.