= Second-order conditions with independent Runge-Kutta abscissae
{title2=$b^Te=1,\quad b^Tc=b^TAe=1/2$}
For a general nonautonomous <Runge-Kutta method>, <Taylor expansion> of stage $i$ gives $k_i=f+h[c_if_t+(Ae)_if_yf]+O(h^2)$. Matching the exact increment $hf+h^2(f_t+f_yf)/2$ gives the three displayed conditions. When internal consistency $c=Ae$ is imposed, the last two coincide. Internal consistency is a usual tableau convention, but is not logically necessary for order two of a generalized tableau.
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