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Mathematics of Computation

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

 
 

 

Numerical analysis of boundary-layer problems in ordinary differential equations


Author: W. D. Murphy
Journal: Math. Comp. 21 (1967), 583-596
MSC: Primary 65.61
DOI: https://doi.org/10.1090/S0025-5718-1967-0225496-9
MathSciNet review: 0225496
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Abstract: We categorize some of the finite-difference methods that can be used to treat the initial-value problem for the boundary-layer differential equation

$\displaystyle (1){\text{ }}\mu y' = f(y,x);y(0) = {y^0}.$

These methods take the form

$\displaystyle (2){\text{ }}\sum\limits_{i = 0}^k {{\alpha _i}{Y_{n + i}} = {h^{... ...amma }}} \sum\limits_{i = 0}^k {{\beta _i}f} ({Y_{n + i}},{x_{n + i}}) + {R_n},$

where $ {\alpha _\nu }$ and $ {\beta _\nu }(\nu = 0,1, \cdots ,k)$ denote real constants which do not depend upon $ n,{R_n}$ is the round-off error, $ \mu = {h^r},0 < \gamma < 1$, and $ h$ is the mesh size. We define a new kind of stability called $ \mu $stability and prove that under certain conditions $ \mu $stability implies convergence of the difference method. We investigate $ \mu $stability and the optimal methods which it allows, i.e., methods of maximum accuracy.

The idea of relating $ \mu $ to $ h$ allows us to study the nature of the difference equation for very small $ \mu $. We can, however, look at this in another way. Given a differential equation in the form of Eq. (1) we ask how can we choose $ h$ so that the associated difference equation will give an accurate approximation. If $ \mu $ is sufficiently small, choose $ h$ by the formula $ h = {\mu ^{1/\gamma }}$ where $ 0 < \gamma < 1$.


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DOI: https://doi.org/10.1090/S0025-5718-1967-0225496-9
Article copyright: © Copyright 1967 American Mathematical Society

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