RK4

is an ODE solver similar to euler’s method, with the iterator taking

the form:

RK-family

method relies on finding k_n iteratively, in RK4 the iteration reads

However,

these relations are

only useful

to derive k2, not from k1 directly, but proxyed through

a complicated y. We now seek to eliminate this proxy. We wonder what

happened to the coefficients of k1 to k4. Thus, we should

taylor-expand k1 to k4, exploiting the aforementioned preserved

relation. Note this expansion has a 2D form if partial derivative is

taken. However we can exploit the principle of total derivative to

simplify the notation.(This is essentially dropping the dependency of

“f” on “y” to achieve a simpler notation).

By

the principle of total derivative:

Take

derivative wrt t:

Exchange

LHS with RHS, time both sides

by

gives:

substitue

( II ) into ( I ) gives:

Note

,in

other words:

Compare

(III) against well-known Taylor expansion of y( t+h ):

We

obtain

*According

to Wikipedia, the full expansion of (III) looks like

Giving

up to:

and

thus