Game Development Reference
In-Depth Information
In this case, the first truncated term, ((∆ t ) 2 / 2 !) v ''( t ), dominates the truncation error,
and this method is said to have an error of order (∆ t ) 2 .
Geometrically, Euler's method approximates a new value, at the current step, for the
function under consideration by extrapolating in the direction of the derivative of the
function at the previous step. This is illustrated in Figure 7-1 .
Figure 7-1. Euler integration step
Figure 7-1 illustrates the truncation error and shows that Euler's method will result in
a polygonal approximation of the smooth function under consideration. Clearly, if you
decrease the step size, you increase the number of polygonal segments and better ap‐
proximate the function. As we said before, though, this isn't always efficient to do since
the number of computations in your simulation will increase and round-off error will
accumulate more rapidly.
To illustrate Euler's method in practice, let's examine the linear equation of motion for
the ship example of Chapter 4 :
T - (C v) = ma
where T is the propeller's thrust, C is a drag coefficient, v is the ship's velocity, m its mass,
and a its acceleration.
Figure 7-2 shows the Euler integration solution, using a 0.5s time step, superimposed
over the exact solution derived in Chapter 4 for the ship's speed over time.
 
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