Inverse linear multistep methods for the numerical solution of initial value problems of ordinary differential equations

Author:
Peter Alfeld

Journal:
Math. Comp. **33** (1979), 111-124

MSC:
Primary 65L05

DOI:
https://doi.org/10.1090/S0025-5718-1979-0514813-6

MathSciNet review:
514813

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Abstract: The well-known explicit linear multistep methods for the numerical solution of ordinary differential equations advance the numerical solution from to by computing some numerical approximation from back values and then evaluating the problem defining function to obtain an approximation of the derivative. In this paper similar methods are proposed that first compute an approximation to the derivative and then compute an approximation to the exact solution, either by evaluating a suitable function, or by solving a nonlinear system of equations. The methods can be applied to initial value problems where the exact solution is explicitly given in terms of the derivative. They can also be applied in the context of the CDS technique for certain stiff initial value problems of ordinary differential equations, introduced in [1 ] and [2]. Local accuracy and stability of the methods are defined and investigated, and specific methods, containing free parameters, are given.

The methods are not convergent, but they yield very good numerical results if applied to the type of problem they are designed for. Their major advantage is that they significantly reduce the amount of implicitness necessary in the numerical solution of certain problems.

**[1]**P. Alfeld and J. D. Lambert,*Correction in the dominant space: a numerical technique for a certain class of stiff initial value problems*, Math. Comp.**31**(1977), no. 140, 922–938. MR**0519719**, https://doi.org/10.1090/S0025-5718-1977-0519719-2**[2]**P. ALFELD,*Correction in the Dominant Space*:*A New Technique for the Numerical Solution of Certain Stiff Initial Value Problems*, Ph.D. Thesis, Univ. of Dundee, Dundee DD1 4HN Scotland, 1977.**[3]**B. L. EHLE,*On Padé Approximations to the Exponential Function and A-Stable Methods for the Numerical Solution of Initial Value Problems*, Univ. of Waterloo, Department of Applied Analysis and Computer Science, Research Report No. CSRR 2010, 1969.**[4]**W. E. ENRIGHT, T. E. HULL & B. LINDBERG, "Comparing numerical methods for stiff systems of o.d.e.s,"*BIT*, v. 15, 1975, pp. 10-48.**[5]**Peter Henrici,*Discrete variable methods in ordinary differential equations*, John Wiley & Sons, Inc., New York-London, 1962. MR**0135729****[6]**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**

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DOI:
https://doi.org/10.1090/S0025-5718-1979-0514813-6

Article copyright:
© Copyright 1979
American Mathematical Society