Implementing second-derivative multistep methods using the Nordsieck polynomial representation

Author:
G. K. Gupta

Journal:
Math. Comp. **32** (1978), 13-18

MSC:
Primary 65L05

DOI:
https://doi.org/10.1090/S0025-5718-1978-0478630-7

MathSciNet review:
0478630

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Abstract | References | Similar Articles | Additional Information

Abstract: A polynomial representation for the second-derivative linear multistep methods for solving ordinary differential equations is presented. This representation leads to an implementation of the second-derivative methods using the Nordsieck polynomial representation. Possible advantages of such an implementation are then discussed.

**[1]**Germund G. Dahlquist,*A special stability problem for linear multistep methods*, Nordisk Tidskr. Informations-Behandling**3**(1963), 27–43. MR**0170477****[2]**W. H. Enright,*Second derivative multistep methods for stiff ordinary differential equations*, SIAM J. Numer. Anal.**11**(1974), 321–331. MR**0351083**, https://doi.org/10.1137/0711029**[3]**W. H. ENRIGHT (1974b), "Optimal second derivative methods for stiff systems," in*Stiff Differential Systems*, R. A. Willoughby (Editor), Plenum Press, New York, pp. 95-109.**[4]**W. H. ENRIGHT, T. E. HULL & B. LINDBERG (1975), "Comparing numerical methods for stiff systems of ODE:s,"*BIT*, v. 15, pp. 10-48.**[5]**C. William Gear,*Numerical initial value problems in ordinary differential equations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR**0315898****[6]**Y. Genin,*An algebraic approach to 𝐴-stable linear multistep-multiderivative integration formulas*, Nordisk Tidskr. Informationsbehandling (BIT)**14**(1974), 382–406. MR**0368438****[7]**G. K. Gupta,*Some new high-order multistep formulae for solving stiff equations*, Math. Comp.**30**(1976), no. 135, 417–432. MR**0423812**, https://doi.org/10.1090/S0025-5718-1976-0423812-1**[8]**Rolf Jeltsch,*Note on 𝐴-stability of multistep multiderivative methods*, Nordisk Tidskr. Informationsbehandling (BIT)**16**(1976), no. 1, 74–78. MR**0411173****[9]**J. D. Lambert,*Computational methods in ordinary differential equations*, John Wiley & Sons, London-New York-Sydney, 1973. Introductory Mathematics for Scientists and Engineers. MR**0423815****[10]**J. D. Lambert and S. T. Sigurdsson,*Multistep methods with variable matrix coefficients*, SIAM J. Numer. Anal.**9**(1972), 715–733. MR**0317548**, https://doi.org/10.1137/0709060**[11]**W. Liniger and F. Odeh,*𝐴-stable, accurate averaging of multistep methods for stiff differential equations*, IBM J. Res. Develop.**16**(1972), 335–348. Mathematics of numerical computation. MR**0345416**, https://doi.org/10.1147/rd.164.0335**[12]**Werner Liniger and Ralph A. Willoughby,*Efficient integration methods for stiff systems of ordinary differential equations*, SIAM J. Numer. Anal.**7**(1970), 47–66. MR**0260181**, https://doi.org/10.1137/0707002**[13]**Matti Mäkelä, Olavi Nevanlinna, and Aarne H. Sipilä,*Exponentially fitted multistep methods by generalized Hermite-Birkhoff interpolation*, Nordisk Tidskr. Informationsbehandling (BIT)**14**(1974), 437–451. MR**0411176****[14]**G. J. Makinson,*Stable high order implicit methods for the numerical solution of systems of differential equations*, Comput. J.**11**(1968/1969), 305–310. MR**0235737**, https://doi.org/10.1093/comjnl/11.3.305**[15]**Arnold Nordsieck,*On numerical integration of ordinary differential equations*, Math. Comp.**16**(1962), 22–49. MR**0136519**, https://doi.org/10.1090/S0025-5718-1962-0136519-5**[16]**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**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1978-0478630-7

Keywords:
Linear multistep methods,
stiff differential equations,
multiderivative methods,
numerical solutions of ordinary differential equations

Article copyright:
© Copyright 1978
American Mathematical Society