Page 47 - Process Modelling and Simulation With Finite Element Methods

34 Process Modelling and Simulation with Finite Element Methods

This assumes that the derivative does not change over the step of size h, which is
only actually true for a linear function. For any function with curvature, this is a
lousy assumption. Consider, for instance, how far wrong we go with a large step
size in Figure 1.2. So clearly, one important point in improving on Euler’s
Method is to be able to use big steps, since it requires small steps for good
accuracy. Euler’s method is called “first order” accurate, as the error only
decreases as the first power of h.
I

4 1-2 I 2 4

Figure 1.2 Curvature effects are lost in the Euler method.
Runge-Kutta methods

So if we want to use big step sizes, we need a “higher order method”, one that
reduces the error faster as step size decreases. A k-th order method has error
which diminishes as hk. Given that it is curvature that we know we are
neglecting, we can estimate the curvature of the graph y(x) by evaluating the
slope f(x) at several intermediate points between x, and xn+1. Second order
accuracy is obtained by using the initial derivative to estimate a point halfway
across the interval, then using the midpoint derivative across the full width of the
interval.

(1.11)

Yn+l - - yn +k, +0(h3)

The upshot is that by making two function evaluations, we have saved a whole
order in accuracy. So, for instance, with a first order method, N calculations
gives us an error O(l/N), but for a second order method, 2N calculations gives
us error O( 1/4N2). It would take N2 calculations to do so well with a first order
method.
Higher order Runge-Kutta methods

Can we do better? Clearly, we can use a three midpoint method to achieve third
order accuracy, a four midpoint method for fourth order accuracy, etc. When