The explicit Euler method is, of course, a one-step method where ϕ(t, y) = f(t, y). In this chapter, our attention focuses mainly on one-step methods. We can now introduce the notion of order of convergence.
Definition — Order of Convergence
A one-step scheme is said to converge with order p if
max1 ≤ n ≤ N | y(tₙ) − yₙ | = O(hᵖ)
where N is the total number of time steps.
The order of convergence of a one-step method depends on the error made at each iteration through the local truncation error, which we now define.
Definition — Local Truncation Error
The local truncation error at the point t = tₙ is defined by:
τₙ₊₁ = [ y(tₙ₊₁) − y(tₙ) ] / h − ϕ(tₙ , y(tₙ))
The local truncation error measures how accurately the exact solution satisfies the difference equation (7.4).
Remark
It is crucial to note that the exact solution y(tₙ) (not the numerical approximation yₙ) is used in the definition of the local truncation error (Equation ). The goal is to measure the error introduced by the difference equation assuming that the method has been exact up to that step.
Remark
The local truncation error alone is not sufficient to determine the convergence order of a one-step method. To draw such a conclusion, one must also introduce the notion of zero-stability (Asher & Petzold, ref. [2]), which lies beyond the objectives of this introductory text.
One can show that if the local truncation error is of order p and the scheme is zero-stable, then the method converges with order p. In what follows, all presented schemes are assumed to be zero-stable so that the order of the local truncation error (Equation 7.6) coincides with the order of convergence in the sense of relation (7.5).