Numerical analysis of boundary-layer problems in ordinary differential equations

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 \[ (1){\text { }}\mu y’ = f(y,x);y(0) = {y^0}.\] These methods take the form \[ (2){\text { }}\sum \limits _{i = 0}^k {{\alpha _i}{Y_{n + i}} = {h^{1 - \gamma }}} \sum \limits _{i = 0}^k {{\beta _i}f} ({Y_{n + i}},{x_{n + i}}) + {R_n},\] 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 $0 < \gamma < 1$.