Abstract
In this paper, a family of fourth orderP-stable methods for solving second order initial value problems is considered. When applied to a nonlinear differential system, all the methods in the family give rise to a nonlinear system which may be solved using a modified Newton method. The classical methods of this type involve at least three (new) function evaluations per iteration (that is, they are 3-stage methods) and most involve using complex arithmetic in factorising their iteration matrix. We derive methods which require only two (new) function evaluations per iteration and for which the iteration matrix is a true real perfect square. This implies that real arithmetic will be used and that at most one real matrix must be factorised at each step. Also we consider various computational aspects such as local error estimation and a strategy for changing the step size.
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References
L. Brusa and L. Nigro,A one-step method for direct integration of structural dynamic equations, Int. J. Num. Meth. in Engng., 15, (1980), 685–699.
J. R. Cash,High order P-stable formulae for the numerical integration of periodic initial value problems, Numer. Math., 37, (1981), 355–370.
J. R. Cash,Efficient P-stable methods for periodic initial value problems, BIT, 24, (1984), 248–252.
M. M. Chawla,Two-step fourth order P-stable methods for second order differential equations, BIT, 21, (1981), 190–193.
M. M. Chawla,Unconditionally stable Numerov-type methods for second order differential equations, BIT, 23, (1983), 541–542.
M. M. Chawla and B. Neta,Families of two-step fourth order P-stable methods for second order differential equations, J. Comp. Appl. Math., 15, (1986), 213–223.
F. Costabile and C. Costabile,Two-step fourth order P-stable methods for second order differential equations, BIT, 22, (1982), 384–386.
I. Gladwell, L. F. Shampine and R. W. Brankin,Automatic selection of the initial step size for an ODE solver, J. Comp. Appl. Math., 18 (1987), 175–192.
I. Gladwell and R. M. Thomas,A-stable methods for second order differential systems and their relation to Padé approximants. In:Rational Approximation and Interpolation. Edited by P. R. Graves-Morris, E. Saff and R. S. Varga. Lecture Notes in Mathematics, 1105, (1984), 419–430, Springer.
I. Gladwell and R. M. Thomas,Efficiency of methods for second order problems, University of Manchester/ UMIST Numerical Analysis Report No. 129. (1987).
E. Hairer,Unconditionally stable methods for second order differential equations, Numer. Math., 32, (1979), 373–379.
N. Houbak, S. P. Nørsett and P. G. Thomsen,Displacement or residual test in the application of implicit methods for stiff problems, I.M.A. J. Numer. Anal., 5, (1985), 297–305.
J. D. Lambert, and I. A. Watson,Symmetric multistep methods for periodic initial value problems, JIMA, 18, (1976), 189–202.
S. P. Nørsett,One step Hermite type methods for numerical integration of stiff systems, BIT, 14, (1974), 63–77.
S. P. Nørsett and P. G. Thomsen,Local error control in SDIRK methods, BIT, 26, (1986), 100–113.
S. P. Nørsett and G. Wanner,The real-pole sandwich for rational approximations and oscillation equations, BIT, 19, (1979), 79–94.
L. F. Shampine,Evaluation of implicit formulas for the solution of ODEs, BIT, 19, (1979), 495–502.
L. F. Shampine,Implementation of implicit formulas for the solution of ODEs, SIAM J. Sci. Stat. Comput., 1, (1980), 103–118.
R. M. Thomas,Phase properties of high order, almost P-stable formulae, BIT, 24, (1984), 225–238.
R. M. Thomas,Efficient fourth order P-stable formulae, University of Manchester/UMIST Numerical Analysis Report No. 103. (1985).
R. M. Thomas,Fourth order P-stable formulae for nonlinear oscillation problems, University of Manchester/UMIST Numerical Analysis Report No. 133. (1987).
H. A. Watts,Starting step size for an ODE solver, J. Comp. Appl. Math., 9, (1983), 177–191.
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Thomas, R.M. Efficient fourth orderP-stable formulae. BIT 27, 599–614 (1987). https://doi.org/10.1007/BF01937279
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DOI: https://doi.org/10.1007/BF01937279