#### CS3414 Afterclass Notes --- 6 June, 2002

Ordinary Differential Equations (parts of Chapters 8 and 12)
1. Introduction

• ODEs: equations involving a unknown functions x(t) and their derivatives.

• Some terminology and basic concepts: IVP, BVP, order of an equation.

• A single IVP:
x'(t) = f(t,x)
x(t0) = x0,
where f() and x0 are given.

• A numerical solution to an IVP consists of a series of points (x0,t0), (x1,t1), ..., (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.