Jackiewicz, Zdzislaw; Renaut, Rosemary Anne; Zennaro, Marino Explicit two-step Runge-Kutta methods. (English) Zbl 0849.65055 Appl. Math., Praha 40, No. 6, 433-456 (1995). This paper analyses the class of explicit two-step Runge-Kutta methods. It is shown that for methods of order 5 or less, these methods require the same number of function evaluations as a Runge-Kutta method of one order less. In particular, a 4-stage order 5 method is constructed and some numerical tests are performed. Reviewer: K.Burrage (Brisbane) Cited in 11 Documents MSC: 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems 65L05 Numerical methods for initial value problems involving ordinary differential equations Keywords:explicit two-step Runge-Kutta methods; 4-stage order 5 method; numerical tests Software:M3RK PDFBibTeX XMLCite \textit{Z. Jackiewicz} et al., Appl. Math., Praha 40, No. 6, 433--456 (1995; Zbl 0849.65055) Full Text: EuDML References: [1] Bellen, A., Jackiewicz, Z., Zennaro, M.: Local error estimation for singly-implicit formulas by two-step Runge-Kutta methods. BIT 32 (1992), 104-117. · Zbl 0783.65062 · doi:10.1007/BF01995111 [2] Burrage, K.: A special family of Runge-Kutta methods for solving stiff differential equations. BIT 18 (1978), 22-41. · Zbl 0384.65034 · doi:10.1007/BF01947741 [3] Burrage, K.: Order properties of implicit multivalue methods for ordinary differential equations. IMA J. Numer. Anal. 8 (1988), 43-69. · Zbl 0637.65066 · doi:10.1093/imanum/8.1.43 [4] Butcher, J. C.: The numerical analysis of ordinary differential equations. Runge-Kutta and general linear methods, New York, John Wiley, 1987. · Zbl 0616.65072 [5] Byrne, G. C., Lambert, R. J.: Pseudo-Runge-Kutta methods involving two points. J. Assoc. Comput. Mach. 13 (1966), 114-123. · Zbl 0135.37802 · doi:10.1145/321312.321321 [6] Hairer, E., Wanner, G.: Multistep-multistage-multiderivative methods for ordinary differential equations. Computing 11 (1973), 287-303. · Zbl 0271.65048 · doi:10.1007/BF02252917 [7] Hairer, E., Wanner G.: On the Butcher group and general multi-value methods. Computing 13 (1974), 1-15. · Zbl 0293.65050 · doi:10.1007/BF02268387 [8] Hull, T. E., Enright, W. M., Fellen, B. M. Sedgwick, A. E.: Comparing numerical methods for ordinary differential equations. SIAM J. Numer. Anal. 9 (1972), 603-637. · Zbl 0221.65115 · doi:10.1137/0709052 [9] Jackiewicz, Z., Renaut, R., Feldstein, A.: Two-step Runge-Kutta methods. SIAM J. Numer. Anal. 28 (1991), 1165-1182. · Zbl 0732.65064 · doi:10.1137/0728062 [10] Jackiewicz, Z., Zennaro, M.: Variable stepsize explicit two-step Runge-Kutta methods. Technical Report No. 125. Arizona State Univ. Math. Comp. vol. 59, 1992, pp. 421-438. · Zbl 0769.65052 · doi:10.2307/2153065 [11] Owren, B., Zennaro, M.: Order barriers for continuous explicit Runge-Kutta methods. Math. Comp. 56 (1991), 645-661. · Zbl 0718.65051 · doi:10.2307/2008399 [12] Renaut, R.: Numerical solution of hyperbolic partial differential equations. Ph.D. thesis. Cambridge University, England, 1985. [13] Renaut, R.: Two-step Runge-Kutta methods and hyperbolic partial differential equations. Math. Comp. 55 (1990), 563-579. · Zbl 0724.65076 · doi:10.2307/2008433 [14] Renaut, R. A.: Runge-Kutta methods for the method of lines solutions of partial differential equations. Submitted (1994.). [15] Rizzi, A. W., Inouye, M.: Time split finite-volume method for three-dimensional blunt-body flow. AIAA J. 11 (1973), no. 11, 1478-1485. · Zbl 0279.76033 · doi:10.2514/3.50614 [16] Verwer, J. G.: Multipoint multistep Runge-Kutta methods I: On a class of two-step methods for parabolic equations. Report NW 30/76. Mathematisch Centrum, Department of Numerical Mathematics, Amsterdam 1976. · Zbl 0332.65043 [17] Verwer, J. G.: Multipoint multistep Runge-Kutta methods II: The construction of a class of stabilized three-step methods for parabolic equations. Report NW 31/76. Mathematisch Centrum, Department of Numerical Mathematics, Amsterdam 1976. · Zbl 0332.65044 [18] Verwer, J. G.: An implementation of a class of stabilized explicit methods for the time integration of parabolic equations. ACM Trans. Math. Software 6 (1980), 188-205. · Zbl 0431.65069 · doi:10.1145/355887.355892 [19] Watt, J. M.: The asymptotic discretization error of a class of methods for solving ordinary differential equations. Proc. Camb. Phil. Soc. 61 (1967), 461-472. · Zbl 0153.18103 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.