Variable stepsize continuous two-step Runge-Kutta methods for ordinary differential equations (Q1815706)

The authors derive sufficient conditions for stability and convergence of two-step continuous Runge-Kutta (TSRK) formulas with variable stepsize, and they present new TSRK formulas with these properties. A two-parameter family of one stage algorithms is derived whose order of convergence is \(p=3\) and stage order of convergence \(q=3\). Values of the parameters sufficient to zero-stability of the formulas are determined. Equations for zero-stable two stage formulas, \(p=4\), \(q=4\), are derived having four parameters. Explicit equations are presented for parameter choices (2.15183,-3.00706,0,2,87785), (.716383,2.17828,.75,2.70983), (.33333,.66666,0,0). A zero-stable three-stage formula, \(p=5\), \(q=5\), is derived having seven parameters. Explicit formulas are presented for a special choice of the parameters. Results obtained from computation applying the algorithm to two test problems are given. The computed orders of convergence are found to agree closely with the designed orders of convergence. When the algorithms are restricted to constant stepsize they coincide with algorithms presented by the authors in an earlier paper [SIAM J. Numer. Anal. 32, No. 5, 1390-1427 (1995; Zbl 0837.65074)].