Abstract
A family of one-step multiderivative methods based on Padé approximants to the exponential function is developed. The methods are extrapolated and analysed for use inPECE mode.
Error constants, stability intervals and stability regions are given in two associated Technical Reports.
Comparisons are made with well-known linear multi-step combinations and combinations using high accuracy Newton-Cotes quadrature formulas as correctors.
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References
O. Axelsson,A class of A-stable methods, BIT 9 (1969), 185–199.
B. L. Ehle,High order A-stable methods for the numerical solution of systems of differential equations, BIT 8 (1968), 276–278.
A. R. Gourlay and J. Ll. Morris,The extrapolation of first order methods for parabolic partial differential equations II, SIAM J. Numer. Anal. 17(5) (1980), 641–655.
J. D. Lambert,Computational Methods in Ordinary Differential Equations, Wiley, Chichester, 1973.
J. D. Lambert and A. R. Mitchell,On the solution of y′=f(x,y) by a class of high accuracy difference formulae of low order, Z. Angew. Math. Phys. 13 (1962), 223–232.
J. D. Lawson and B. L. Ehle,Asymptotic error estimation for one-step methods based on quadrature, Aeq. Math. 5 (1970), 236–246.
J. D. Lawson and J. Ll. Morris,The extrapolation of first order methods for parabolic partial differential equations I, SIAM J. Numer. Anal. 15(6) (1978), 1212–1224.
B. Lindberg,On smoothing and extrapolation for the trapezoidal rule, BIT 11 (1971), 29–52.
W. E. Milne,A note on the numerical integration of differential equations, J. Res. Nat. Bur. Standards 43 (1949), 537–542.
M. H. Padé,Sur la répresentation approchée d'une fonction par des fractions rationelles, Ann. de l'École Normale Supérieure, Vol. 9 (Suppl.), 1892.
E. H. Twizell and A. Q. M. Khaliq,One step multiderivative methods for first order ordinary differential equations, Brunel University Department of Mathematics Technical Report TR/02/81, 1981.
E. H. Twizell and A. Q. M. Khaliq,Stability regions for one step multiderivative methods, Brunel University Department of Mathematics, Technical Report TR/04/81, 1981.
E. H. Twizell and P. Smith,Numerical modelling of heat flow in the human torso I: finite difference methods, Proceedings of POLYMODEL4: diffusion convection problems, Sunderland Polytechnic, May 1981.
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Twizell, E.H., Khaliq, A.Q.M. One-step multiderivative methods for first order ordinary differential equations. BIT 21, 518–527 (1981). https://doi.org/10.1007/BF01932848
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DOI: https://doi.org/10.1007/BF01932848