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Exponentially Fitted Multistep Integration Algorithms for Ordinary Differential Equations

IP.com Disclosure Number: IPCOM000079138D
Original Publication Date: 1973-May-01
Included in the Prior Art Database: 2005-Feb-26
Document File: 6 page(s) / 155K

Publishing Venue

IBM

Related People

Liniger, W: AUTHOR [+2]

Abstract

The integration algorithm for ordinary differential equations represented by the flow charts is based on a family of predictor and corrector formulas (Image Omitted) respectively, depending on the step number parameter k. It is described in detail in [1] and is designed for an accurate and efficient numerical solution of systems, of an arbitrary number N of first order ordinary differential equations, of the form y' = -Dy + Phi(x,y) (3) particularly in the case where such a system is stiff. In equations (1), (2) and (3), y and M represent vectors of N components, the quantities G(i), G*(i), Q and D are diagonal matrices of order N, h is the integration step, and i = 0,...,k, represents the ith backward difference.

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Exponentially Fitted Multistep Integration Algorithms for Ordinary Differential Equations

The integration algorithm for ordinary differential equations represented by the flow charts is based on a family of predictor and corrector formulas

(Image Omitted)

respectively, depending on the step number parameter k. It is described in detail in [1] and is designed for an accurate and efficient numerical solution of systems, of an arbitrary number N of first order ordinary differential equations, of the form y' = -Dy + Phi(x,y) (3) particularly in the case where such a system is stiff. In equations
(1), (2) and (3), y and M represent vectors of N components, the quantities G(i), G*(i), Q and D are diagonal matrices of order N, h is the integration step, and i = 0,...,k, represents the ith backward difference. The formulas (1) and (2) are exponentially fitted [2] to the system (3), i.e., the coefficients G(i) are chosen to depend on the parameter Q = hD via the formula

(Image Omitted)

and similarly for G/*/(i) which depends on Q via coefficients Psi/*/(ij) and Xi*(ij) The quantities Psi(ij), Xi(ij), Psi*(ij) and Xi*(ij) in turn are expressible as simple functions of certain auxiliary quantities Gamma(ij) and Gamma*(ij).

The first part of the algorithm, represented by flow charts 1 and 2, is an efficient way of computing the auxiliary quantities Gamma(ij) and Gamma*(ij), respectively. This is done recursively with respect to i. For small values of q = hd, where q and d represent diagonal elements of Q and D, respectively, the diagonal elements g(i) and g*(i) of G(i9 and G*(i), respectively, are computed via the series expansions given in the Appendix, rather than via closed expressions such as equation (4), to avoid rounding error difficulties.

The second part of the algorithm, represented by flow chart 3, is a predictor-modifier-corrector-modifier (PMCM) and a predictor-modifier-iterated-corrector (PMIC) method. Flow chart 3 involves the following steps: FLOW CHART #3
1) READ IN INPUT:

Initial point of the interval XI

End point of the interval XN

Step-size (integration step) H

Stopping criterion for iteration EPS

Initial conditions YINIT(I)

Diagonal entries D(I)

Print interval IPRIN(I)

Indicator for PMCM vs. PMIC IND

1

Page 2 of 6

2) RECORD INPUT

i.e.: XI, XN, H, EPS, YINIT(I), D(I), IPRIN(I).

IND.
3) If the product of H*D(I) is less than 0.0085 then use series
3. expansion to compute coefficients, otherwise use closed form

for the computation.
4) Compute starting values by Runge Kutta method (4th order) o...