In this derivation of the 2nd order Runge-Kutta method it's said that from a certain point $y$'s dependence on $t$ in $f(t,y)$ will be neglected. Why is this possible?
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I guess 'neglect' is too strong for what is actually being done... They are just saying that they are using Taylor's approximations of $f$, as a function of $t$ and $y$, meaning that they are explaing the increments in $f$ by increments in $t$ and $y$. This approximation holds regardless of the fact that $y$ is actually a function of $t$. – PierreCarre Feb 25 '20 at 19:51
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$\newcommand{\by}{\mathbf{y}}$ This is slightly wrong, as there are indeed terms missing in the Taylor expansion. These terms follow the pattern of the $\by$-derivatives, so that no information is lost, but ...
They should have just said that to derive the order conditions resp. the method of the highest possible order for the number of stages they only consider autonomous systems $\by'=f(\by)$. One has to take care that $\by$ is a vector and $f$ is vector valued, so that $f'(\by)$ is a matrix, the Jacobi matrix. Higher derivatives then are tensors, vector-valued symmetric multi-linear forms.
This is general enough, as you can transform any non-autonomous system into an artificially autonomous one as $$ \frac{d}{dt}\pmatrix{t\\\by}=F((t,\by))=\pmatrix{1\\f(t,\by)} $$ where now the state vector has the time $t$ as additional component.
Lutz Lehmann
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