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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Recursive collocation for the numerical solution of stiff ordinary differential equations
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by H. Brunner PDF
Math. Comp. 28 (1974), 475-481 Request permission

Abstract:

The exact solution of a given stiff system of nonlinear (homogeneous) ordinary differential equations on a given interval I is approximated, on each subinterval ${\sigma _k}$ corresponding to a partition ${\pi _N}$ of I, by a linear combination ${U_k}(x)$ of exponential functions. The function ${U_k}(x)$ will involve only the "significant" eigenvalues (in a sense to be made precise) of the approximate Jacobian for ${\sigma _k}$. The unknown vectors in ${U_k}(x)$ are computed recursively by requiring that ${U_k}(x)$ satisfy the given system at certain suitable points in ${\sigma _k}$ (collocation), with the additional condition that the collection of these functions $\{ {U_k}\}$ represent a continuous function on I satisfying the given initial conditions.
References
    G. Bjurel et al., Survey of Stiff Ordinary Differential Equations, Report NA 70.11, Dept. of Computer Science, Royal Institute of Technology, Stockholm, 1970.
  • 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
  • Wolfgang Hahn, Stability of motion, Die Grundlehren der mathematischen Wissenschaften, Band 138, Springer-Verlag New York, Inc., New York, 1967. Translated from the German manuscript by Arne P. Baartz. MR 0223668
  • 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
  • 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 260181, DOI 10.1137/0707002
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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Math. Comp. 28 (1974), 475-481
  • MSC: Primary 65L05
  • DOI: https://doi.org/10.1090/S0025-5718-1974-0347089-9
  • MathSciNet review: 0347089