Linear multistep methods with mildly varying coefficients

Author:
J. D. Lambert

Journal:
Math. Comp. **24** (1970), 81-93

MSC:
Primary 65.61

DOI:
https://doi.org/10.1090/S0025-5718-1970-0280010-7

MathSciNet review:
0280010

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Consideration of a common assumption in the theory of weak stability of linear multistep methods for ordinary differential equations leads to the study of a class of linear multistep methods with mildly varying coefficients. It is well known that, in the case of constant-coefficient methods, optimal stable methods suffer from weak instability. Corresponding methods of the new class, of step-number and , which do not suffer from weak instability are derived.

**[1]**G. Dahlquist, "Convergence and stability in the numerical integration of ordinary differential equations,"*Math. Scand.*, v. 4, 1956, pp. 33-53. MR**18**, 338. MR**0080998 (18:338d)****[2]**P. Henrici,*Discrete Variable Methods in Ordinary Differential Equations*, Wiley, New York, 1962. MR**24**#B1772. MR**0135729 (24:B1772)****[3]**T. E. Hull & A. C. R. Newbery, "Corrector formulas for multi-step integration methods,"*J. Soc. Indust. Appl. Math.*, v. 10, 1962, pp. 351-369. MR**27**#2130. MR**0152150 (27:2130)****[4]**A. Ralston, "Relative stability in the numerical solution of ordinary differential equations,"*SIAM Rev.*, v. 7, 1965, pp. 114-125. MR**31**#2831. MR**0178574 (31:2831)****[5]**R. W. Hamming, "Stable predictor-corrector methods for ordinary differential equations,"*J. Assoc. Comput. Mach.*, v. 6, 1959, pp. 37-47. MR**21**#973. MR**0102179 (21:973)****[6]**H. Brunner, "Stabilization of optimal difference operators," Z.*Angew. Math. Phys.*, v. 18, 1967, pp. 438-444. MR**36**#1111. MR**0218022 (36:1111)****[7]**G. Dahlquist, "A special stability problem for linear multistep methods,"*Nordisk Tidskr. Informations-Behandling*, v. 3, 1963, pp. 27-43. MR**30**#715. MR**0170477 (30:715)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65.61

Retrieve articles in all journals with MSC: 65.61

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0280010-7

Keywords:
Linear multistep methods,
weak stability,
initial value problem,
variable coefficient linear difference equations

Article copyright:
© Copyright 1970
American Mathematical Society