CS3414 Afterclass Notes --- 6 June, 2002

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), ..., (x_N,t_N), where t_N is the `final' time for which the solution is desired, and x_k is my approximation to x(t_k).

   • 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 t_final 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: x_k - x(t_k)

       • Local error: the error made at step k, assuming the answer was correct at step k-1,
         i.e., x_k - z(t_k), where z(t) exactly solves the IVP:
           z'(t) = f(t,x)
           z(t_k-1) = x_k-1.