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Consider the following example. Suppose that our o.d.e. is

(14) 
where , subject to the boundary condition

(15) 
Of course, we can solve this problem analytically to give

(16) 
Note that the solution is a monotonically decreasing function of .
We can also solve this problem numerically using Euler's method. Appropriate
gridpoints are

(17) 
where
. Euler's method yields

(18) 
Note one curious fact. If then
.
In other words, if the steplength is made too large then the numerical
solution becomes an oscillatory function of of
monotonically increasing amplitude:
i.e., the numerical solution diverges from the actual
solution. This type of catastrophic failure of a numerical integration
scheme is called a numerical instability. All simple integration
schemes become unstable if the steplength is made sufficiently large.
Next: RungeKutta methods
Up: Integration of ODEs
Previous: Numerical errors
Richard Fitzpatrick
20060329