Page 100 - Standard Handbook Of Petroleum & Natural Gas Engineering

Numerical Methods 89

dY
y(0) = 0 and -(O) = U
dx
U is unknown and must be chosen so that y(L) = 0. The equation may be solved
as an initial value problem with predetermined step sizes so that xn will equal
L at the end point. Since y(L) is a function of U, it will be denoted as y,(U)
and an appropriate value of U sought so that

Any standard root-seeking method that does not utilize explicitly the derivative
of the function may be employed.
Given two estimates of the root U,, and U,, two solutions of the initial value
problem are calculated, yL(Uoo) and yL(U,), a new estimate of U is obtained where

and the process is continued to convergence.
There are three basic classes of second-order partial differential equations
involving two independent variables:

1. Parabolic

2. Elliptic

3. Hyperbolic

where @ = $(x,y,u,au/ax,au/ay). Each class requires a different numerical
approach. (For higher-order equations and equations in three or more variables,
the extensions are usually straightforward.)
Given a parabolic equation of the form