Two-step Runge-Kutta: Theory and practice (Q1592700)

The paper is concerned with the development of adaptive two-step Runge-Kutta methods for non-stiff initial value problems. These methods have the following structure \[ y_{n+1} = \eta y_{n-1} + (1-\eta) y_n + h \sum_{i=1}^s \left(v_i f(Y_i^{(n-1)}) + w_i f(Y_i^{(n)}) \right) \] \[ Y_i^{(n)} = u_i y_{n-1} + (1-u_i) y_n + h \sum_{j=1}^s \left(a_{ij} f(Y_j^{(n-1)}) + b_{ij} f(Y_j^{(n)}) \right) \] where \( \eta, u_i, v_i, w_i, a_{ij}\) and \(b_{ij} \) are given real values for \( i, j = 1, 2, \dots, s \). The second set of equations define implicitly \( Y_i^{(n)} \) for \( i = 1, 2, \dots , s \), which are approximations to \( y(x_n + c_i h) \) for suitable real values \( c_i \). The first set of equations, called the advancing formulae, provide the approximate solution at the new grid-point and the remaining equations are the stage equations to be used on a particular step. The B-series theory [cf. \textit{E. Hairer, S. P. Nørsett} and \textit{G. Wanner}, Solving ordinary differential equations. I: Nonstiff problems (1987; Zbl 0638.65058), pp. 242-248] is applied to study local and global errors with both constant and variable step-sizes. An embedded pair of methods is constructed and compared with the RK5(4)6M pair of \textit{J. R. Dormand} and \textit{P. J. Price} [J. Comput. Appl. Math., 6, 19-26 (1980; Zbl 0448.65045)] on the test problems taken from the DETEST [cf. \textit{T. E. Hull, W. H. Enright, B. M. Fellen}, and \textit{A. E. Sedgwick}, SIAM J. Numer. Anal. 9, 603-637 (1972; Zbl 0221.65115)].