CS3414 Afterclass Notes --- 6 June, 2002
Ordinary Differential Equations (parts of Chapters 8 and 12)
- ODEs: equations involving a unknown functions x(t) and their
- Some terminology and basic concepts: IVP, BVP, order of an equation.
- A single IVP:
x'(t) = f(t,x)
x(t0) = x0,
f() and x0 are given.
- A numerical solution to an IVP consists of a series of points
(xN,tN), where tN is the
`final' time for which the solution is desired, and
xk is my approximation to x(tk).
- Some important themes in numerical ODEs:
- Step size selection: in general, a smaller step size means
better accuracy; a larger step size means less work, i.e., we get to
tfinal with fewer function evaluations.
- Adaptive step size selection based on local error estimates,
which are themselves (typically) based on comparing a good
solution and a `better' solution.
- Local vs. global error:
- Global error: xk - x(tk)
- Local error: the error made at step k, assuming the answer
was correct at step k-1,
i.e., xk - z(tk),
where z(t) exactly solves the IVP:
z'(t) = f(t,x)
z(tk-1) = xk-1.