Game Development Reference

In-Depth Information

16.4.2 Taylor Series Method

Any method for approximating the solution to a differential equation by taking

one step at a time assumes the form

(

)

(

)

(

)

y xh yx Fxy

+=

+

,

,

(16.64)

i

i

i

i

where the function
F
is some function that produces an approximation to the de-

rivative of
y
over the interval

]

. For Euler's method, the function
F
is

simply the function
f
. To find a function
F
that achieves greater accuracy than

that provided by Euler's method, we consider the Taylor series (see Appendix D)

of

,

xh

+

[

x

i

i

(

)

y xh

+

:

i

2

3

h

h

()

(

) ()

′

()

′′

()

3

()

y xh yx yx

+=

+

+

yx

+

y x

+

.

(16.65)

i

i

i

i

i

2!

3!

For a differential equation written in the form of Equation (16.58), the derivatives

of

()

y x
can all be calculated using the relationship

()

n

()

(

n

−

1

)

(

)

y

xf

=

xy

,

.

(16.66)

By taking

−

1

derivatives, we can calculate the Taylor series approximation of

k

(

)

y xh

+

to
k
-th order in the step size
h
, yielding

i

h

2

h

k

(

)

(

)

(

)

(

)

(

)

y xh y f

+≈+

xy

,

+

f

′

xy

,

++

f

k

−

1

xy

,

,

(16.67)

i

i

i

i

i

i

i

i

2!

k

!

()

where

, this reduces to Euler's method. Writing Equation

(16.67) in the form of Equation (16.64), we have

y

=

yx

. When

=

1

k

i

i

(

)

(

)

(

)

y xh yx Txy

+=

+

,

,

(16.68)

i

i

k

i

i

(

)

where

xy
is defined as

,

T

k

i

i

2

k

−

1

h

h

(

)

(

)

′

(

)

(

k

−

1

)

(

)

T

xy

,

=

fxy

,

+

f xy

,

+

+

f

xy

,

. (16.69)

k

i

i

i

i

i

i

i

i

2!

k

!

This is known as the
k-th order Taylor series method
.

Since
y
is a function of
x
, we must be careful to evaluate the total derivatives

(

)

of

xy
in the Taylor series. The first derivative is

,

f

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