A class of -stable advanced multistep methods

Authors:
Jack Williams and Frank de Hoog

Journal:
Math. Comp. **28** (1974), 163-177

MSC:
Primary 65L05

MathSciNet review:
0356519

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Abstract: A class of *A*-stable advanced multistep methods is derived for the numerical solution of initial value problems in ordinary differential equations. The methods, of all orders of accuracy up to ten, only require values of *y'* and are self starting. Results for the asymptotic behaviour of the discretization error and for estimating local truncation error are also obtained. The practical implementation of the fourth order method is described and the method applied to some stiff equations. Numerical comparisons are made with Gear's method.

**[1]**Owe Axelsson,*A class of 𝐴-stable methods*, Nordisk Tidskr. Informationsbehandling (BIT)**9**(1969), 185–199. MR**0255059****[2]**F. H. Chipman,*𝐴-stable Runge-Kutta processes*, Nordisk Tidskr. Informationsbehandling (BIT)**11**(1971), 384–388. MR**0295582****[3]**Germund G. Dahlquist,*A special stability problem for linear multistep methods*, Nordisk Tidskr. Informations-Behandling**3**(1963), 27–43. MR**0170477****[4]**G. Dahlquist et al.,*Survey of Stiff Ordinary Differential Equations*, The Royal Institute of Technology, Stockholm, Report NA 70.11, 1970.**[5]**James W. Daniel,*Nonlinear equations arising in deferred correction of initial value problems*, Acta Ci. Venezolana**19**(1968), 123–128 (English, with Spanish summary). MR**0255062****[6]**James W. Daniel, Victor Pereyra, and Larry L. Schumaker,*Iterated deferred corrections for initial value problems*, Acta Ci. Venezolana**19**(1968), 128–135 (English, with Spanish summary). MR**0255063****[7]**Byron L. Ehle,*High order 𝐴-stable methods for the numerical solution of systems of D.E.’s*, Nordisk Tidskr. Informationsbehandling (BIT)**8**(1968), 276–278. MR**0239762****[8]**C. W. Gear,*The automatic integration of stiff ordinary differential equations.*, Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968) North-Holland, Amsterdam, 1969, pp. 187–193. MR**0260180****[9]**C. W. Gear, "DIFSUB for solution of ordinary differential equations,"*Comm. ACM*, v. 14, 1971, pp. 185-190.**[10]**C. William Gear,*Numerical initial value problems in ordinary differential equations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR**0315898****[11]**Peter Henrici,*Discrete variable methods in ordinary differential equations*, John Wiley & Sons, Inc., New York-London, 1962. MR**0135729****[12]**Bernie L. Hulme,*Discrete Galerkin and related one-step methods for ordinary differential equations*, Math. Comp.**26**(1972), 881–891. MR**0315899**, 10.1090/S0025-5718-1972-0315899-8**[13]**F. T. Krogh,*On Testing a Subroutine for the Numerical Integration of Ordinary Differential Equations*, Jet Propulsion Laboratory, Pasadena, Calif., Tech. Mem. No. 217, 1970.**[14]**Leon Lapidus and John H. Seinfeld,*Numerical solution of ordinary differential equations*, Mathematics in Science and Engineering, Vol. 74, Academic Press, New York-London, 1971. MR**0281355****[15]**Anthony Ralston,*A first course in numerical analysis*, McGraw-Hill Book Co., New York-Toronto-London, 1965. MR**0191070****[16]**J. Barkley Rosser,*A Runge-Kutta for all seasons*, SIAM Rev.**9**(1967), 417–452. MR**0219242****[17]**L. F. Shampine and H. A. Watts,*Block implicit one-step methods*, Math. Comp.**23**(1969), 731–740. MR**0264854**, 10.1090/S0025-5718-1969-0264854-5**[18]**H. A. Watts and L. F. Shampine,*𝐴-stable block implicit one-step methods*, Nordisk Tidskr. Informationsbehandling (BIT)**12**(1972), 252–266. MR**0307483****[19]**H. A. Watts,*A-Stable Block Implicit One-Step Methods*, Sandia Laboratories Report SC-RR-71 0296.**[20]**K. Wright,*Some relationships between implicit Runge-Kutta, collocation Lanczos 𝜏 methods, and their stability properties*, Nordisk Tidskr. Informationsbehandling (BIT)**10**(1970), 217–227. MR**0266439**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0356519-8

Keywords:
*A*-stable,
advanced multistep methods,
stiff systems

Article copyright:
© Copyright 1974
American Mathematical Society