Some new multistep methods for solving ordinary differential equations

Authors:
G. K. Gupta and C. S. Wallace

Journal:
Math. Comp. **29** (1975), 489-500

MSC:
Primary 65L05

MathSciNet review:
0373290

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Abstract | References | Similar Articles | Additional Information

Abstract: Three sets of linear multistep formulae for solving stiff and nonstiff ordinary differential equations are presented. Two of the sets are based on Adams-Moulton and stiff formulae used by Gear [1969]. A third set of formulae based on least-squares approximation is shown to be stiffly-stable up to order 8 and is suitable for solving stiff differential equations.

**[G]**Germund Dahlquist,*Convergence and stability in the numerical integration of ordinary differential equations*, Math. Scand.**4**(1956), 33–53. MR**0080998****[G]**Germund G. Dahlquist,*A special stability problem for linear multistep methods*, Nordisk Tidskr. Informations-Behandling**3**(1963), 27–43. MR**0170477****[C]**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****[C]**C. W. Gear,*The automatic integration of ordinary differential equations*, Comm. ACM**14**(1971), no. 3, 176–179. MR**0388778****[C]**C. W. Gear,*The automatic integration of ordinary differential equations*, Comm. ACM**14**(1971), no. 3, 176–179. MR**0388778****[G]**K. GUPTA & C. S. WALLACE (1974), "A note about the stability and truncation error of linear multistep methods,"*SIAM J. Numer. Anal.*(Submitted.)**[P]**Peter Henrici,*Discrete variable methods in ordinary differential equations*, John Wiley & Sons, Inc., New York-London, 1962. MR**0135729****[N]**MORRISON (1969),*Introduction to Sequential Smoothing and Prediction*, McGraw-Hill, New York.**[A]**Arnold Nordsieck,*On numerical integration of ordinary differential equations*, Math. Comp.**16**(1962), 22–49. MR**0136519**, 10.1090/S0025-5718-1962-0136519-5**[C]**C. S. Wallace and G. K. Gupta,*General linear multistep methods to solve ordinary differential equations*, Austral. Comput. J.**5**(1973), 62–69. MR**0362919****[O]**Olof B. Widlund,*A note on unconditionally stable linear multistep methods*, Nordisk Tidskr. Informations-Behandling**7**(1967), 65–70. MR**0215533**

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1975-0373290-5

Keywords:
Linear multistep methods,
stiff differential equations,
predictor-corrector methods,
numerical solution of ordinary differential equations

Article copyright:
© Copyright 1975
American Mathematical Society