Adaptive Control of One-Step Methods

Abstract

In the previous chapter we met a number of one-step methods that allowed us to approximate the solution x ∈ C¹ ([t₀, T], R^d) of the initial value problem

$$x'\, = \,f\left( {t,\,x} \right),\,\,\,\,\,\,\,x\left( {t_0 } \right)\, = \,x_0,$$

by a mesh function xΔ, for any given mesh Δ = {t₀, t₁,…, t _n }, t ₀ < t₁ < … < t _n = T. For methods of order p we obtained a characterization of the discretization error in the form

$$\mathop {\max }\limits_{t\, \in \,\Delta } \left| {x\left( t \right)\, - \,x_\Delta \left( t \right)} \right|\, \le \,C\tau _\Delta ^p,$$

where τΔ denotes the maximal step size of the mesh Δ. But this information alone does not permit us to specify a priori a suitable mesh Δ. In fact, without further information about the solution, we can hardly come up with a better idea than an equidistant mesh

$$\Delta _n \, = \,\left\{ {t_0 \, + \,j{{T\, - \,t_0 } \over n}:\,j\, = \,0,\,1, \ldots,\,n} \right\},$$

with some τ = τΔ. However, it cannot possibly be expected that such equidistant meshes will do justice to the multitude of different problems. Accordingly, in this chapter we address the question of how to construct efficiently a problem-adapted mesh.