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A collocation method chooses an approximate solution from a finite-dimensional trial space and forces the residual of a differential equation to vanish at selected nodes. Collocation Runge-Kutta methods apply this principle to polynomial approximations on each time step.

Given distinct nodes $c_i$, a collocation Runge-Kutta method uses the Lagrange basis ${\ell }_j$ with $a_{ij}={\int }_0^{c_i}{\ell }_j(\tau )d\tau$ and $b_j={\int }_0^1{\ell }_j(\tau )d\tau$. Its stages enforce the differential equation at the collocation nodes.

This applies the collocation method to the time-step polynomial of an ordinary differential equation.