Linear multistep methods with mildly varying coefficients

Author:
J. D. Lambert

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

MSC:
Primary 65.61

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]**Germund Dahlquist,*Convergence and stability in the numerical integration of ordinary differential equations*, Math. Scand.**4**(1956), 33–53. MR**0080998****[2]**Peter Henrici,*Discrete variable methods in ordinary differential equations*, John Wiley & Sons, Inc., New York-London, 1962. MR**0135729****[3]**T. E. Hull and A. C. R. Newbery,*Corrector formulas for multi-step integration methods*, J. Soc. Indust. Appl. Math.**10**(1962), 351–369. MR**0152150****[4]**Anthony Ralston,*Relative stability in the numerical solution of ordinary differential equations*, SIAM Rev.**7**(1965), 114–125. MR**0178574****[5]**R. W. Hamming,*Stable predictor-corrector methods for ordinary differential equations.*, J. Assoc. Comput. Mach.**6**(1959), 37–47. MR**0102179****[6]**Hermann Brunner,*Stabilization of optimal difference operators*, Z. Angew. Math. Phys.**18**(1967), 438–444 (English, with German summary). MR**0218022****[7]**Germund G. Dahlquist,*A special stability problem for linear multistep methods*, Nordisk Tidskr. Informations-Behandling**3**(1963), 27–43. MR**0170477**

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