Second Order Accurate Methods

1.5.3 Second Order Accurate Methods

In step by step computations we have to distinguish between the local error, the error that is committed at a single step, and the global error, the error of the final results. Recall that we say that a method is accurate of order p if its global error is approximately proportional to h p . Euler’s method is only first order accurate; we shall present a method that is second order accurate. To achieve the same accuracy as with Euler’s method the number of steps can then be reduced to about the square root of the number of steps in Euler’s method. In √ the above ball problem this means 1500 ≈ 40 steps. Since the amount of work is closely proportional to the number of steps this is an enormous savings!

Another question is how the step size h is to be chosen. It can be shown that even for rather simple examples (see below) it is adequate to use very different step sizes in different parts of the computation. Hence the automatic control of the step size (also called adaptive control) is an important issue.

60 Chapter 1. Principles of Numerical Calculations Both requests can be met by an improvement of the Euler method (due to Runge 18 )

obtained by applying the Richardson extrapolation in every second step. This is different from our previous application of the Richardson idea. We first introduce a better notation by writing a system of differential equations and the initial conditions in vector form

dy/dt = f(t, y), y(a) = c,

(1.5.9) where y is a column vector that contains all the state variables. 19 With this notation methods

for large systems of differential equations can be described as easily as methods for a single equation. The change of a system with time can then be thought of as a motion of the state vector in a multidimensional space, where the differential equation defines the velocity field. This is our first example of the central role of vectors and matrices in modern computing.

For the ball example, we have a = 0, and by (1.5.5) and (1.5.6),  y 1   x 

−g − zy 4 sin φ where

cR 2

(y 3 ) 2 + (y 4 ) 2 .

The computations in the step which leads from t n to t n +1 are then as follows:

i. One Euler step of length h yields the estimate:

y n ∗ +1 =y n + hf(t n ,y n ).

ii. Two Euler steps of length 1 2 h yield another estimate:

hf (t 2 n 2 +1/2 ,y n +1/2 ), where t n +1/2 =t n + h/2.

y n +1/2 =y n +

hf (t n ,y n ),

y ∗∗ n +1 =y n +1/2 +

iii. Then y n +1 is obtained by Richardson extrapolation:

y n +1 =y n ∗∗ +1 + (y ∗∗ n +1 −y ∗ n +1 ).

It is conceivable that this yields a second order accurate method. It is left as an exercise (Problem 1.5.2) to verify that this scheme is identical to the following somewhat simpler scheme known as Runge’s second order method:

18 Carle David Tolmé Runge (1856–1927), German mathematician. Runge was professor of applied mathematics at Göttingen from 1904 until his death.

19 The boldface notation is temporarily used for vectors in this section only, not in the rest of the book.

61 where we have replaced h by h n in order to include the use of variable step size. Another

1.5. Numerical Solution of Differential Equations

explanation of the second order accuracy of this method is that the displacement k 2 equals the product of the step size and a sufficiently accurate estimate of the velocity at the midpoint of the time step. Sometimes this method is called the improved Euler method or Heun’s method, but these names are also used to denote other second order accurate methods.