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## Numerical instabilities

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 grid-points are (17)

where . Euler's method yields (18)

Note one curious fact. If then . In other words, if the step-length 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 step-length is made sufficiently large.   Next: Runge-Kutta methods Up: Integration of ODEs Previous: Numerical errors
Richard Fitzpatrick 2006-03-29