Order of a method

The order of a method is determined by applying the method to the example equation.

The value of is determined for the method and compared with the Taylor expansion of

For example, the Euler method gives

,

The expressions agree to within terms of O(t) and we call the Euler method a first order method. It looks like the errors go as t² , but they really go as t.

To see this, suppose the problem is

and f(t,y) satisfies the Lipschitz condition and the second derivative is bounded.

Let yⁿ be the numerical solution at tⁿ and y_ex(tⁿ) be the exact solution at tⁿ. The numerical solution at the next time step, tⁿ⁺¹, is

(1)

The exact solution at tⁿ⁺¹ has the following Taylor series.

This is also

⁽²⁾

Subtract Eq. (1) from Eq. (2).

Define the error as

and rewrite the equations.

Rearrange and take absolute values of both sides.

Use the Lipschitz constant and the maximum of the second derivative.

The following equation is also true using the triangular inequality.

Thus

This equation shows that the error of the Euler method at the n+1-st time is the error at the n-th time plus a term which is proportional to t. Thus the error is O(t) and we say the method is first order. This analysis need not be repeated for each method, since it can be found by comparing with the Taylor expansion of exp(lt) as shown above.

Take Home Message: The order of a method can be found by either a detailed mathematical analysis, or by comparing the method to the Taylor expansion of an exponential. The last method is easier and faster.