# A partial derivation of the RK4 ODE-solver // RK4积分器的部分推导

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 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 