Variable-stepsize explicit two-step Runge-Kutta methods

Authors:
Z. Jackiewicz and M. Zennaro

Journal:
Math. Comp. **59** (1992), 421-438

MSC:
Primary 65L06

MathSciNet review:
1136222

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Abstract: Variable-step explicit two-step Runge-Kutta methods for the numerical solution of ordinary differential equations are studied. Order conditions are derived and the results about the minimal number of stages required to attain a given order are established up to order five. The existence of embedded pairs of continuous Runge-Kutta methods and two-step Runge-Kutta methods of order and *p* is proved. This makes it possible to estimate local discretization error of continuous Runge-Kutta methods without any extra evaluations of the right-hand side of the differential equation. An algorithm to construct such embedded pairs is described, and examples of (3, 4) and (4, 5) pairs are presented. Numerical experiments illustrate that local error estimation of continuous Runge-Kutta methods based on two-step Runge-Kutta methods appears to be almost as reliable as error estimation by Richardson extrapolation, at the same time being much more efficient.

**[1]**Kevin Burrage,*Order properties of implicit multivalue methods for ordinary differential equations*, IMA J. Numer. Anal.**8**(1988), no. 1, 43–69. MR**967843**, 10.1093/imanum/8.1.43**[2]**J. C. Butcher,*The numerical analysis of ordinary differential equations*, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1987. Runge\mhy Kutta and general linear methods. MR**878564****[3]**George D. Byrne and Robert J. Lambert,*Pseudo-Runge-Kutta methods involving two points*, J. Assoc. Comput. Mach.**13**(1966), 114–123. MR**0185823****[4]**E. Hairer and G. Wanner,*Multistep-multistage-multiderivative methods of ordinary differential equations*, Computing (Arch. Elektron. Rechnen)**11**(1973), no. 3, 287–303 (English, with German summary). MR**0378422****[5]**E. Hairer, S. P. Nørsett, and G. Wanner,*Solving ordinary differential equations. I*, Springer Series in Computational Mathematics, vol. 8, Springer-Verlag, Berlin, 1987. Nonstiff problems. MR**868663****[6]**P. J. van der Houwen,*Construction of integration formulas for initial value problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. North-Holland Series in Applied Mathematics and Mechanics, Vol. 19. MR**0519726****[7]**P. J. van der Houwen and B. P. Sommeijer,*On the internal stability of explicit, 𝑚-stage Runge-Kutta methods for large 𝑚-values*, Z. Angew. Math. Mech.**60**(1980), no. 10, 479–485 (English, with German and Russian summaries). MR**614285**, 10.1002/zamm.19800601005**[8]**P. J. van der Houwen and B. P. Sommeijer,*A special class of multistep Runge-Kutta methods with extended real stability interval*, IMA J. Numer. Anal.**2**(1982), no. 2, 183–209. MR**668592**, 10.1093/imanum/2.2.183**[9]**T. E. Hull, W. H. Enright, B. M. Fellen, and A. E. Sedgwick,*Comparing numerical methods for ordinary differential equations*, SIAM J. Numer. Anal.**9**(1972), 603–637; errata, ibid. 11 (1974), 681. MR**0351086****[10]**Z. Jackiewicz, R. Renaut, and A. Feldstein,*Two-step Runge-Kutta methods*, SIAM J. Numer. Anal.**28**(1991), no. 4, 1165–1182. MR**1111459**, 10.1137/0728062**[11]**Z. Jackiewicz and M. Zennaro,*Explicit two-step Runge-Kutta methods*(submitted).**[12]**-,*Variable stepsize explicit two-step Runge-Kutta methods*, Tech. Rep. 125, Department of Mathematics, Arizona State University, Tempe, 1990.**[13]**B. Owren,*Continuous explicit Runge-Kutta methods with applications to ordinary and delay differential equations*, Ph.D. thesis, Norges Tekniske Høgskole, Institutt for Matematiske Fag, Trondheim, Norway, 1989.**[14]**B. Owren and M. Zennaro,*Continuous explicit Runge-Kutta methods*, Proc. London 1989 Conference on Computational ODE's.**[15]**Brynjulf Owren and Marino Zennaro,*Order barriers for continuous explicit Runge-Kutta methods*, Math. Comp.**56**(1991), no. 194, 645–661. MR**1068811**, 10.1090/S0025-5718-1991-1068811-2**[16]**R. A. Renaut,*Two-step Runge-Kutta methods and hyperbolic partial differential equations*, Math. Comp.**55**(1990), no. 192, 563–579. MR**1035943**, 10.1090/S0025-5718-1990-1035943-3**[17]**R. A. Renaut-Williamson,*Numerical solution of hyperbolic partial differential equations*, Ph.D. thesis, Cambridge University, England, 1985.**[18]**J. G. Verwer,*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.**[19]**-,*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.**[20]**-,*An implementation of a class of stabilized explicit methods for the time integration of parabolic equations*, ACM Trans. Math. Software**6**(1980), 188-205.**[21]**M. Zennaro,*Natural continuous extensions of Runge-Kutta methods*, Math. Comp.**46**(1986), no. 173, 119–133. MR**815835**, 10.1090/S0025-5718-1986-0815835-1

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DOI:
https://doi.org/10.1090/S0025-5718-1992-1136222-8

Article copyright:
© Copyright 1992
American Mathematical Society